Differential Geometry of Spray and Finsler Spaces 9789048156733, 9789401597272, 9048156734

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Differential Geometry of Spray and Finsler Spaces

Differential Geometry of Spray and Finsler Spaces by

Zhongmin Shen Department of Mathematical Sciences, Indiana University-Purdue University at Indianapolis, Indianapolis, IN, U.S.A.

SPRINGER-SCIENCE+BUSINESS MEDIA, B.V.

A C.I.P. Catalogue record for this book is available from the Library of Congress.

ISBN 978-90-481-5673-3 ISBN 978-94-015-9727-2 (eBook) DOI 10.1007/978-94-015-9727-2

Printed on acid-free paper

All Rights Reserved © 2001 Springer Science+Business Media Dordrecht Originally published by Kluwer Academic Publishers in 2001 Softcover reprint of the hardcover I st edition 200 I No part of the material protected by this copyright notice may be reproduced or utilized in any form or by any means, electronic or mechanical, including photocopying, recording or by any information storage and retrieval system, without written permission from the copyright owner.

Contents Introduction

1

1 Minkowski Spaces 1.1 Minkowski Functionals 1.2 Cartan Torsion . 1.3 Varga Equation .

3 4 11 18

2 Finsler Spaces 2.1 Finsler Metrics 2.2 Spherical Metrics . 2.3 Hyperbolic Metrics

21

3

35 35

21 30 32

SODEs and Variational Problems 3.1 SODEs . . . . . . . . . . . . . .. . .. . . . .. . 3.2 Variational Problems . .. .. . . . . .. . .. . . 3.3 R elationship Between Euler-Lagrange Equations

40 43

4 Spray Spaces 4.1 Sprays . . . .. . . . . . . 4.2 Finsler Sprays . . . .. . . 4.3 Geodesics of Funk Metrics

47 47

5 S-Curvature 5.1 Volumes . 5.2 S-Curvature .. 5.3 Geodesic Flows

63

6 Non-Riemannian Quantities 6.1 Berwald Curvature of Sprays . .. .. . 6.2 Landsberg Curvature of Finsler Metrics 6.3 Finsler Surfaces . . . . . . . . . . . . .

77 77 84 89

7 Connections 7.1 Berwald Connection of Sprays . 7.2 Chern Connection of Finsler Metrics . 7.3 Covariant Derivatives Along Geodesics

55 59 63

68 74

.

95 95

99

101

CONTENTS

Vl

8

Riemann Curvature 8.1 Riemann Curvature of Sprays . . . . . . . . 8.2 Riemann Curvature of Finsler Metrics .. . 8.3 Riemann Curvature of Riemannian Metrics 8.4 Geodesic Fields and Riemann Curvature 8.5 Variational Formulas . . . . . . . . .. .. .

107 107 117 126 129 131

9

Structure Equations of Sprays 9.1 Berwald Connection Forms . . . . . . . 9.2 Curvature Forms and Bianchi Identities 9.3 R-Quadratic and R-Flat Sprays . . . ..

133 133 135 137

10 Structure Equations of Finster Metrics 10.1 Ricci Identities for g . . . . . . . . . . . 10.2 Ricci Identities for C and L . . . . . . . 10.3 R-Quadratic and R-Flat Finsler Metrics

143 143 147 148

11 Finster Spaces of Scalar Curvature 11 .1 Finsler Metric of Scalar Curvatures . 11.2 Finsler Metrics of Constant Curvature . .. 11.3 Riemannian Spaces of Constant Curvature .

153 153 160 167

12 Projective Geometry 12.1 Projectively Related Sprays . . . . . 12.2 Projectively Related Finsler Metrics 12.3 Projectively Related Einstein Metrics 12.4 Inverse Problems . . . . . . . . . . .

173 173 180 186 190

. . . .

13 Douglas Curvature and Weyl Curvature 13.1 Douglas Curvature of Sprays . . . . . . . 13.2 Projectively Affine Sprays and Douglas Metrics 13.3 Weyl Curvature of Sprays . . . . . . . . . . . . 13.4 Isotropic Sprays and Finsler Metrics of Scalar Curvature . 13.5 Projectively Flat Sprays and Finsler Metrics . 13.6 Berwald-Weyl Curvature . . . . . . . . . . . .

197 197 199 204 205 207 212

14 Exponential Maps 14.1 Exponential Map of Sprays . . . . . . . . 14.2 Jacobi Fields in Spray Spaces . . . . . . . 14.3 Comparison Theorems for Finsler Spaces . 14.4 Jacobi Fields in Isotropic Spray Spaces . 14.5 Finsler Spaces of Constant Curvature

221 221 225 228 231 238

Bibliography

243

Index

255

Acknowledgments

This book is an expanded version of the lecture notes given by the author during his visits to Taiwan (May, 1998) , Mainland China (June, 1998) and Brazil (May, 1999). The author would like to thank Professors Jyh-Yang Wu, Yi-Bing Shen and Keti Tenenblat for their warm hospitality. This research project was supported in part by the Center for Theoretical Sciences in Taiwan, the Natural Science Foundation of China and the CNPq in Brazil. The author thanks many mathematicians including, P. Antonelli, S. Bacs6, M. Berger, M. Matsumoto, H. Shimada, C. Udri§te, etc., for their valuable comments and great help. He is grateful to S. S. Chern for his encouragement and support throughout this research project . Finally, the author would like to thank Chang-Wan Kim and Xinyue Chen for carefully reading through the primary version of his book and correcting several mistakes. Finally, the author wishes to take this opportunity to thank his wife, Tianping, for her support and understanding during the preparation of this manuscript.

Zhongmin Shen Department of Mathematical Sciences Indiana University Purdue University Indianapolis 402 N. Blackford Street Indianapolis, Indiana 46202-3216 USA [email protected]

Introduction In this book we study sprays and Finsler metrics. Roughly speaking, a spray on a manifold consists of compatible systems of second-order ordinary differential equations. A Finsler metric on a manifold is a family of norms in tangent spaces, which vary smoothly with the base point. Every Finsler metric determines a spray by its systems of geodesic equations. Thus, Finsler spaces can be viewed as special spray spaces. On the other hand, every Finsler metric defines a distance function by the length of minimial curves. Thus Finsler spaces can be viewed as regular metric spaces. Riemannian spaces are special regular metric spaces. In 1854, B. Riemann introduced the Riemann curvature for Riemannian spaces in his ground-breaking Habilitationsvortrag. Thereafter the geometry of these special regular metric spaces is named after him. Riemann also mentioned general regular metric spaces, but he thought that there were nothing new in the general case. In fact, it is technically much more difficult to deal with general regular metric spaces. For more than half century, t here had been no essential progress in this direction until P. Finsler did his pioneering work in 1918. Finsler studied the variational problems of curves and surfaces in general regular metric spaces. Some difficult problems were solved by him. Since then, such regular metric spaces are called Finsler spaces. Finsler, however, did not go any further to introduce curvatures for regular metric spaces. He switched his research direction to set theory shortly after his graduation. It was L. Berwald who first successfully extended the notion of Riemann curvature to Finsler spaces. He also introduced a notion of non-Riemannian quantity- the Berwald curvature [Bw1]. From this point of view, Berwald is the founder of differential geometry of Finsler spaces. J. Douglas extended the Berwald curvature to general sprays. He also extended H. Weyl's projective invariant for affine connections to general sprays [Dg1]. Local geometric structures of sprays and Finsler metrics have been understood in a great depth after P. Finsler's pioneering work in 1918 (see [Fi] [Ca] [Ru] [Bu3] [Ma4], etc). The geometric methods developed in Finsler geometry are useful for studying some problems arising from biology, physics and other fields (see [AbPa] [AnZa1] [AnZa2] [AninMa] [As1] [Bj] [Mi1J[MiAn], etc). However, due to complicated tensor computations , spray geometry as well as Finsler geometry has made many beginners turn away from this subject. Indeed, one can easily get lost in the deep forest of tensors. In this book , we t ry to define quantities in an index-free form and derive equations using tensors only when necessary. We introduce several quantities of sprays or Finsler metrics such as the Riemann curvature, Berwald curvature and Landsberg curvature, etc. The names of these quantities are given for the first time, although these quantities have appeared in literatures. In Chapter 1, we discuss some basic properties of Minkowski functionals and introduce the notion of Cartan torsion for Minkowski functionals. In particular, we show how a two-dimensional Minkowski functional depends on the main scalar (the core part of the Cartan torsion in dimension two). In Chapter 2,

1

2

INTRODUCTION

we define Finsler metrics and discuss many important examples . In Chapter 3, we show that the geodesics of a Finsler metric satisfy a system of secondorder ordinary differential equations, that are the Euler-Lagrange equations of the variation of arc-length. We also discuss variational problems of first order and second-order ordinary differential equations. In Chapter 4, we introduce the notion of sprays, based on our discussion in the previous chapters. The spray induced by a Finsler metric is called a Finsler spray. By studying general sprays, we would have a better understanding of Finsler metrics. In Chapters 5 and 6, we introduce several non-Riemannian quantities for sprays and Finsler metrics, which always vanish for Riemannian metrics. We calculate these quantities for some interesting sprays and Finsler metrics. We raise an open problem whether or not there are Landsberg metrics which are not of Berwald type. In Chapter 7, the concept of connection is introduced. Using the canonical connection, we define the parallelism along geodesics. In Chapter 8, we introduce the most important quantity - Riemann curvature for sprays and Finsler metrics. It is defined as the Jacobi endomorphism of the variation of geodesics. In Chapters 9 and 10, we derive several important equations for the quantities introduced in the previous chapters. The flag curvature of Finsler metrics is the generalization of sectional curvature of Riemannian metrics. It is a function of a tangent plane and a vector (pole vector) in the plane. We say that a Finsler metric is of scalar curvature if the flag curvature is independent of the tangent planes containing the pole vector. When the flag curvature is constant, the Finsler metric is said to be of constant curvature. In Chapter 11, we prove several theorems for Finsler metrics of scalar curvature and constant curvature. Some interesting metrics of constant curvature are discussed as well. In Chapter 12, we discuss Projective Geometry of spray and Finsler metrics. We prove the Rapcsak theorem which is one of the most important theorems in Projective Geometry. We derive an equation for the Riemann curvature under a projective change. In Chapter 13, we continue the discussion on projective geometry, and introduce two essential projective invariants, the Douglas curvature and the Weyl curvature. These two quantities play important roles in understanding the projective properties of sprays and Finsler metrics. In Chapter 14, we study some global problems via the exponential map and Jacobi fields. We prove several comparison theorems for positive definite Finsler spaces and describe isotropic sprays and Finsler metrics of constant curvature. The topics in this book are selected based on author's own interests. Some important topics and applications such as Lagrange geometry are only briefly discussed. The interested readers should consult books written by R. Miron and others (Mi1)[MiAn].

Chapter 1

Minkow ski Spaces In this chapter, Minkowski spaces will be introduced. First, let us take a look at Euclidean spaces. Let Rn denote then-dimensional canonical real vector space. Define

IYI :=

n

L 1Yil

(1.1)

2,

i=l

1·1 is called the standard Euclidean norm on Rn. We denote JRn = (Rn, I ·I) which is called the standard Euclidean space. The standard inner product (-,-) on R n is defined by n

(u, v) :=

L uivi,

u = (ui), v =(vi)

ERn.

i=l

This gives rise to the Euclidean norm

IYI = .J(Y:0. More general, consider a nondegenerate bilinear symmetric form Q on an n-dimensional vector space V. Take a basis {ei} i=I for V and express Q as follows (1.2)

where u = uiei, v = viei and (aij) is a symmetric matrix with det (aij) -1- 0. This gives rise to a nondegenerate quadratic form on V

L(y)

:=

Q(y, y).

(1.3)

Q is called an inner product in V if it is positive definite. In this case, o:(y) = JQ(y, y) is called an Euclidean norm on V and the pair (V, o:) is called an Euclidean space. Every Euclidean space is linearly isometric to the standard Euclidean space JRn .

3

CHAPTER 1. MINKOWSKI SPACES

4

1.1

Minkowski Functionals

We can generalize nondegenerate quadratic forms. Definition 1.1.1 Let V be a finite dimensional vector space. A Minkowski functional on V is a function L : V --+ R which has the following properties:

(i) Lis

c=

on V \ {0},

(iia) L is positively homogeneous of degree two, i.e.,

L(>..y) = >.. 2 L(y),

>.. > 0, y E V

(1.4)

(iib) for every 0 =j:. y E V, the fundamental form gy on V is nondegenerate , where

gy(u,v)

1 ()2

:=

2osot [L(y + su +tv)] is=t=O·

(1.5)

A Minkowski space is a pair (V, L) where V is a finite dimensional vector space and L is a Minkowski functional on V.

By definition, if Q is a nondegenerate symmetric bilinear form on a vector space V, then L(y) := Q(y,y) is a Minkowski functional. For a Minkowski functional L on a vector space V, the indicatrix of L

s = s+ us_ is defined by S± := { y E V, L(y) = ±1 } ·

(1.6)

If L changes sign on V, then S is disconnected. In this case, the cone C of L is non-empty.

C := { y E V, L(y) = 0} =j:. 0.

(1.7)

For a vector y E S, the tangent space of S at y, TyS , can be identified with a hyperplane W y in TxM in a natural way, where

Wy

:={wE V,

gy(y,w)

= o}.

(1.8)

1.1. MINKOWSKI FUNCTIONALS

5

Example 1.1.1 Define Le : R 2 -+ R by

where E ::j:. 0. LE is a Minkowski functional on R 2 . The indicatrix of L_ 1 is the hyperbola in R 2 and that of L 1 is the unit circle in R 2 . U Let V be an n-dimensional vector space and {ei} f= 1 a basis for V. Then L = L(yiei) is a function of (yi) E Rn. Put

g;j(y) Then

1 ()2£ := -~(y) . 2 uy'uyJ

(1.9)

gy(u,v) = gij(y)uivi,

where u = uiei and v = viei. It follows from the homogeneity of L that 18L

.

2 ()y i (y) = gij(y)yl

(1.10)

= %(y)yiyi.

(1.11)

L(y)

Observe that the sign of the determinant det(%(Y)) does not change on V \ {0} . We can define if det(gi3·) > 0 1 ind(L):= { _ 1 if det(gij) < 0. ind (L) is called the index of L .

(1.12)

CHAPTER 1. MINKOWSKI SPACES

6

Let k(y) denote the number of negative eigenvalues of 9v· By continuity, we see that k = k(y) is independent of y E V \ {0} and the index of Lis given by ind (L) = (-l)k. If k

= 0, it follows from

(1.11) that

L(y) > 0,

y E V \ {0}.

In this case, L is said to be positive definite and (V, L) is called a positive definite Minkowski space. Usually, we denote a positive definite Minkowski functional by

F(y)

:=

JL(Y).

The reader should consult A.C. Thompson (Tho) for comprehensive discussion on positive definite Minkowski functionals. Example 1.1.2 Define F : Rn-+ [O,oo) by

F(y)

:=

where ,\ is an arbitrary nonnegative number. One can easily verify that F is a positive definite Minkowski functional on Rn. ~

Example 1.1.3 There are many Minkowski functionals on a real vector space V in the following form

F(y)

:=

1

[Q(y, y, y,y)j4

where Q = Q(u,v,w,z) is a symmetric multi-linear form on V. Consider the following functional on R 2 I

F(u,v) := {u 4 +2cu 2 v 2 +v 4 }

4

.

< c < 3, then F is a positive definite Minkowski functional on R 2 . K. Okubo studied the following type of functionals

If 0

(1.13)

where A, J-L are positive numbers. Rewriting it as follows

1.1. MINKOWSKI FUNCTIONALS one can see that if A and

f..L

3- 2v'2 =

7

satisfy A

1

v'2 < - hab(ae] = 0 4>av 1 - [24>hab + 4>a4>b](b = 0. Contracting (1.23) with

~a

(1.22) (1.23)

yields

4>d~dv 1 - [4>d~d¢b(b

+ 24>hab~a(b]

(1.24)

= 0.

Adding (1.24) to (1.22) yields (1.25) Plugging it into (1.23) gives hab(b = 0. From (1.25), we see that (=f. 0, otherwise v 1 = 0, then va = -4>(a + ~av 1 = 0. This proves (1.21). Thus det(hab(~)) = 0. Q.E.D.

Example 1.1.7 Let

where det ( Aab)

=f. 0.

4>: Rn- 1

-t R be given by

Define

·- (Ay1y1 L(y).-

+ AabYaYb)z . + AaY1Ya 1

By Lemma 1.1.3, we know that for y

y

= (1, ~)

det (9iJ(Y))

E Rn with 4>(0

=f. 0.

=f. 0,

1.2. CARTAN TORSION

11

Thus L is a singular Minkowski functional. L is called a K ropina functional on R n. A general Kropina functional on R n is a singular Minkowski functional in the following form

L(y ) = (L a(Y) ) 2 (J(y) where L 0 is a singular Minkowski functional and (3 is a 1-form on Rn. Let L(O = L(e, · · ·, ~n) be a Minkowski functional on Rn-l and A, A 2 , ··· ,An be constants. Define

Then L is a singular Minkowski functional on R n.

1. 2

Cart an Torsion

For a singular Minkowski functional L on a vector space V with regular conical domain C, there is an important quantity introduced by P. Finsler [Fi) and emphased by E. Cartan [Ca). Fix a basis {e;}f= 1 for V. L = L(yiei) becomes a function on an open cone in R n. For a vector y E C \ {0}, let

cijk(y)

:=

1

4Ly'yiyk(y).

(1.26)

Define Cy : V ® V ® V -+ R by

Cy(u ,v,w)

.

.

= Cijk(y)u'v1 w

k

,

where u = uiei, v = vi ej and w = wkek. Cy is a multi-linear symmetric form on V. It follows from the homogeneity of L that

C y(y, v , w) = 0.

(1.27)

The family C := {Cy}yEC\{O} is called the Cartan torsion. The following proposition is trivial, due to E. Cartan. Proposition 1.2.1 ([Ca)) For a Minkowski functional L on a vector space V, C = 0 if and only if L is quadratic.

For a vector y E C \ {0}, define ly : V-+ R by n

ly(u)

:=

L ij= l

Cy(u, ei, e1)gii(y) ,

(1.28)

CHAPTER 1. MINKOWSKI SPACES

12

where {ei}f= 1 be an arbitrary basis for V and 9iJ(Y) := gy(e;,ej)· We call the family I := {ly }yEC\{O} the mean Cartan torsion. The mean Cartan torsion is an important quantity for Minkowski functionals. Let l;(y) := ly(e;). It is easy to verify the following (1.29) Thus I= 0 if and only if det(9ij(y)) =constant. There are many non-quadratic Minkowski functionals with I = 0. For example, the Berwald-Moor functional in (1.14) has vanishing mean Cartan torsion. It is proved by A. Deike that every positive definite Minkowski functional with I = 0 must be Euclidean [De]. In [JiSh], we show that there are infinitely many non-Euclidean singular Minkowski functionals with I= 0 which are defined on a small open cone in Rn. For a positive definite Minkowski functional L on a vector space V, define

\\Cyll

:=

sup

9y(v ,v)=1

\\C\1

Jcy(v,v,v)j,

:=

Note that C = 0 if and only if \\C\1 = 0. Thus Riemannian features of L at certain degree.

sup

L(y)=1

\\C\1

\\Cy\\.

measures the non-

Let a(y) = Ja;jyiyJ be an Euclidean norm on a vector space V and fJ(y) b;yi a linear functional on V. Consider

F(y)

:=

a(y)

+ fJ(y).

F or L = F 2 is called a Randers functional.

Assume that

\\/3\\o: := sup \fJ(y) \ =

j aiJ b;bj < 1,

o:(y )=1

=

where (aii) := (a;j)- 1 . Then it is easy to see that F is a positive definite Minkowski functional. By an elementary argument, (1.30) See Example 1.2.2 or [BaChSh1] for a proof in dimension two. See also [Sh8] for a proof in higher dimensions. A Randers functional is a linear shift of an Euclidean norm. In the "universe" of Minkowski functionals ("planets" ), Randers functionals are the "planets" closest to the "Euclidean system". No wonder, their Cartan torsions have uniform bound.

1.2. CARTAN TORSION

13

Rander space

Rand r space

•

•

Euclidean system

•

Randers space

Randers space

Minkowski Planes: For a (singular) Minkowski functional L on an oriented vector plane V, the Cartan torsion is determined by a scalar function on V \ {0}. First we prove the following Lemma 1.2.2 Let (V,L) be a Minkowski plane. For a vector y E V with L(y) =f. 0, there is a vector yj_ E V \ {0} such that

(1.31)

where c = ind (L) denotes the index of L defined in (1 .12}. Moreover, {y, yj_} is positively oriented if L(y) > 0 and negatively oriented if L(y) < 0.

0

Proof of Lemma 1.2.2: Take an oriented basis {e 1 ,e 2 } for V. This basis determines a global coordinate system (u, v) in V. Let L(u, v) := L(ue 1

+ vez).

14

CHAPTER 1. MINKOWSKI SPACES

For a vector y = ue 1

+ ve 2

E V \ {0}, define y.l by

(1.32) Observe that

LvLuu - LuLuv = [ LuuLvv - LuvLuv] V

+ LuLvv = [LuuLvv - LuvLuv] U.

- LvLuv Thus

Lv[LvLuu- LuLuv] + Lu[-LvLuv (uLu + vLv)[LuuLvv- LuvLuv] 2L[LuuLvv- LuvLuv]·

+ LuLvv]

Using these identities, one can easily verify that

gy(y, y.l) gy (y.l' y.l)

Thus y.l satisfies (1.31).

Q.E.D.

The pair {y, y.l} is called the Berwald frame at y. Define I( ) ·= Cy(y.l , y .L ,y.L) =I ( .L) y

L(y)

.

Y

y

(1.33)

.

From the definition , we see that I has the following homogeneity property:

I(.\y)

= I(y),

.\ > 0,

y E V \ {0}.

We call I the main scalar of L . Assume that L is a positive definite functional on a vector plane V. Hence

As in the proof of Lemma 1.2.2, we fix a basis { e1 , e 2 } for V and denote by (u , v) the corresponding global coordinate system. Then L(u,v) = L(ue 1 +ve 2 ) is a function on the uv-plane. By the homogeneity of L, we have

Luuu = -

(~) 3 Lvvv,

Luuv =

(~) 2 Lvvv,

Luvv = -

(~)Lvvv·

1.2. CARTAN TORSION

+ vez

Observe that for y = ue 1 I(y)

=

+ 3Luuv( -Lv) 2 Lu + 3Luvv( -Lv)(Lu) 2 + Lvvv(Lu) 3

1 Luuu( -Lv) 3

4

1 u3 (

15

~Lv + Lu

4 L ( 2LLvv -

3

r

L ( LuuLvv - LuvLuv) 2

Lvvv

~

LvLv)

2£ 2 Lvvv 3 •

(2LLvv- LvLv)

Thus

2

I( y ) -_

2£ 2 Lvvv

(1.34)

3.

( 2LLvv - LvLv) 2

Express Las

[u¢(~)

L(u, v) =

where¢= ¢(0 is a positive c= function with

r,

¢c. 0. Then for y = e1 +~e2 (1.35)

Example 1.2.1 Let · - v2+2a _ [ 2- - u U a

L(u,v) .- -

(v)l+a]2 , U

u > 0, v > 0,

where a > 0. Plugging ¢ = ~l+a into (1.35) gives

I=

1 + 2a

Ja(1 +a)

=constant.

Note that mini=2, a>O

Example 1.2.2 Let

maxi= oo. a>O

CHAPTER 1. MINKOWSKI SPACES

16

where IBI < 1 and E = ± (the sign of u). Plugging ¢ = ~ + EB into (1.35), we obtain the main scalar at y = (1, ~)

I = -~ 2(

cB~ · ) 1/4 I y~+cB l+e

It is easy to see that

Clearly, the above estimate holds for any two-dimensional Randers functional + /3 with B := ll/3lla < 1. Note that in dimension two,

F = a

IICII =max III. Thus

IICII ~

~J1- J1- B2 .

(1.36)

The bound (1.36) is actually true for Randers functionals in any dimension. One can prove this using dimension reduction method. The proof is left to the U reader. Let L = F 2 be a positive definite Minkowski functional on V = span{ e1 , e 2 }. Parameterize C := F - 1 (1) by c(s) such that

gc(s)(c(s), c(s)) = 1. Let

I (s)

:=

I(c(s)).

According to 0 . Varga [Va], the vector-valued function c(s) satisfies the following equation (1.37) c"(s) + I(s)c'(s) + c(s) = 0. See Section 1.3 below for a proof. Below we are going to express Minkowski funct ionals for a prescribed main scalar I(s) . Let ¢(s) and '1/J(s) be two linearly independent solutions of the following equation (1.38) y"(s) + I(s)y'(s) + y(s) = 0. Put

+ 'ljJ(s)e2. Assume that '1/J(s) f. 0 and x(s) c(s)

Then c(s) satisfies (1.37). inverse on an interval. Define

:=

¢(s)e1

:= ¢(s)/'I/J(s) has the (1.39)

17

1.2. CARTAN TORSION Let I denote the range of x and

C := {ue1 +vez E V, ; E I, F(ue1 +vez) >

o}.

F is a positive function on C. Lemma 1.2.3 The function F defined in (1.39) is a positive definite Minkowski functional on C and the main scalar ofF is I( s). Proof Clearly, F is positively homogeneous of degree one. Let ue 1 + vez E C

satisfy

F(ue

1

+ ve 2 )

= 1 and s =

Since

x(s)

x- 1 (;) .

Then

¢(s)

u

= 1/;(s) = ;)

we obtain v = 1/;(s).

This implies ue 1 + ve 2 = c(s). Therefore F- 1 (1) is the curve c. One can prove that F is a positive definite Minkowski functional on C and the main scalar of Q.E.D. Fat c(s) is just I(s). The proof is left to the reader. Now we give the list of all singular positive definite Minkowski planes with constant main scalar. This is due to [Bw5]. We divide the problem into three cases. Case 1: I 2

> 4. The general solution of (1.37) is given by (1.40)

where a and b are linear independent constant vectors. The Minkowski functional F is given by (1.41) where fJ1, fJz E V*. Case 2: I 2

= 4.

The general solution of (1.37) is given by (1.42)

where a and b are linear independent constant vectors. The Minkowski functional F is given by I fJ1(y)} (1.43) F(y) = fJz(y) exp { 2fJz(y) .

CHAPTER 1. MINKOWSKI SPACES

18 Case 3: 12

< 4. The general solution of (1.37) is given by (1.44)

where a and b are linear independent constant vectors. The Minkowski functional F is given by (1.45)

The formulas of Minkowski functionals with constant main scalar are also given in [MaSh1][MaSh2].

1.3

Varga Equation

In the previous section, we mentioned the Varga equation (1.37) for two-dimensional Minkowski functionals. In this section, we will prove the Varga equation for Minkowski functionals in all dimensions. Readers who are not familiar with geodesics and the Levi-Civita connection in Riemannian geometry can skip this section. Every Minkowski functional L on a vector space V induces a Riemannian metric g on V \ {0} as follows. Define [Jy(u, v) := gy(u, v),

u,v E TyV:::::: V.

(1.46)

The indicatrix S = L - l (1) U L - l ( -1) of L is a hypersurface in V.

s

For a point y E S, we can identify TyS with the following hyperplane W y in V Wy := { v E V, gy(y,v) =

o.}.

1.3. VARGA EQUATION

19

Define ilv : TyS x TyS--+ R by (1.4 7)

ilv(u,v) = gy(u,v) ,

g := {gy }yEs is a Riemannian metric on S. Define C y : TyS 0 TyS --+ TyS by gy ( Cy(u, v),

w) := Cy(u, v, w),

(1.48)

C = { Cy }yES is called the Cartan tensor on S. Let

.(x, y)v 2 )} 4 ,

(2.19)

where >.(x,y) and fL(x,y) are positive functions. As shown in Example 1.1.3, Okubo's metrics can be expressed in the form (2.18). Thus, they are positive definite Finsler metrics if >.(x, y) and fl(X, y) satisfy ;n

3 - 2v.::

1

>..(x,y)

1

;n

= 3 + 2v;n2 < -(-) < ;n = 3 + 2v 2. 11 x, Y 3- 2v 2

We also come across lots of singular Finsler metrics in applications. Let L : T M --+ R be a function satisfying (2.1). Let CM C T M \ {0} be an open cone, i.e., CM = UxEMCxM is a fiber bundle over M such that each fiber CxM is an open cone in TxM \ {0}. Suppose that L has the following properties (a) Lis coo in CM, (b) the fundamental form gy is nondegenerate for y E CM, then Lis called an singular Finsler metric and (M, L) is called a singular Finsler space. The Cartan torsion can be defined for singular Finsler metrics exactly in the same way as for the regular ones. Example 2.1.4 Given an arbitrary function I(s), a< s o

o:(y)

x-1 (o:(vl) f3(y)

'

yETM.

(2.21)

Clearly, F satisfies the homogeneity condition

F(>.y)

= >.F(y),

>. > 0.

Let {a, b} be a frame on M, which is dual to {o:, /3}. At each point x E M, let

Cx(s)

:=

(s)ax

+ 1/J(s)bx.

Cx is a curve in TxM. Observe that (s)

4> o

x- 1(M) 1/J(s)

- 1

- ·

(2.22)

Thus the indicatrix of Fx in TxM is the curve Cx. Lemma 1.2.3 tells us that Fx is a singular positive definite Minkowski functional on TxM with the main scalar I(s). We obtain a Finsler metric F = {Fx }xEM on M such that the main scalar of Fx in TxM is the same I(s) for all x E M . It is known t hat when I(s) =constant, F is a Berwald metric. This fact is first proved by L. Berwald

d

~w~.

Example 2.1.5 Let m be an odd integer. The following metric is a generalized Berwald-Moor metric. I

F(y)

:=

[ai1 ···i=(x)yi 1 ···yi=]m,

(2.23)

where ai 1 ···i= (x) are functions on an open subset D c Rn. F is a singular Finsler metric on D in many cases. The Finsler metric in (2.23) is also called an m-th root metric. In his work on generalized Berwald spaces [Wa1] [Wa2], V.V. Wagner first studied a special class of m-th root metrics. Later on, M. Matsumoto and others wrote several papers on m-th root metrics. See [Ma12], [MaOk] and references therein. d

Example 2.1.6 Given>.> 0, a, b, c > 0. Define L: TR2 --+ [0, oo) by

(2.24)

28

CHAPTER 2. FINSLER SPACES

where .A, o:i are positive constants. Lis a B erwald metric. It is locally Minkowskian if and only if v3 = 0. See Examples 6.1.2 and 6.1.2 below. This Finsler metric has an application to heterochrony [AnHaMo] . There is another interesting metric studied by Antonelli [Anl] .

L

:=

exp [4

a:: : 2

·(ax+ by)] exp [2:

~: tan- 1 CD] (u 2 + v2 ).

(2.25)

This metric is actually a locally Minkowskian metric. See Example 8.2.5 below.

u

The above mentioned Finsler metrics belong to the following family of singular Finsler metrics on R 2 . L

= e(x,y) N(u, v),

(2.26)

where N is a Minkowski functional with constant main scalar. The Finsler metric in (2 .26) was used in coral reef ecology [AnlnMa]. By [Lal][La2][Bw5], we know that F is a Berwald metric. A Finsler metric L is called an Antonelli metric if at every point, there is a local coordinate system (called an adapted system) in which qk(y) = qk(y 1 , · · · yn) are functions of (yi) only. This kind of Finsler metrics were first studied by P.L. Antonelli [An2]. He calls them y-Berwald metrics. Antonelli metrics arise in time-sequencing change models in the evolution of colonial systems. Any two-dimensional Antonelli metric is conformal to a locally Minkowskian metric which can be expressed in an adapted coordinate system as follows L = eax+by+c N(u, v), (2.27) where Fa is a Minkowski functional on R 2 . Conversely, any Finsler metric in the form (2.27) must be an Antonelli metric (see Example 4.2.3). This fact was proved in (AnlnMa] . Two-dimensional Antonelli metrics were also studied in (AnMaZa]. The simplest Antonelli metrics are those with constant Christoffel symbols in adapted coordinates. They are also called constant-Berwald metrics. ConstantBerwald metrics play an important role in the recent development of ecology. Two-dimensional constant-Berwald metrics must be locally projectively flat . In fact, they are either locally Minkowskian or pseudo-Riemannian. See [AnHaMo] [AnMa2] [AninMa] for further discussion. Example 2.1. 7 ([In]) In Luneburg's theory of binocular vision, the binocular visual space is presented in the form of a three-dimensional Riemannian space of constant negative curvature with the following metric on the ball $ 3 ( r) of radius r = 2/ Ffi

2.1. FINSLER METRICS

29

Note that 4L 1 is the standard Poincare metric on the unit ballllli 3 (1) when "'= -4. Taking a consideration of other facts , R. Ingarden generalizes Luneburg's metric to ( u2 + v2 + w2) 2 (2.28) La:= 2 · [1 +

t"' (x2 + y2 + z2) J (u2 + v2 + a2v2)

Clearly, La is conformally Minkowskian. Some Finsler metrics are constructed from Lagrange metrics. We now briefly discuss Lagrange metrics below. Further discussion will be given in the next chapter on variational problems. Definition 2.1.2 Let I= (a, /3) and N be an (n -!)-dimensional manifold. A function ¢ : I x TN -+ R is a called a Lagrange metric on I x N if it has the following properties

(i) ¢is coo on I x (TN\ {0}), (ii) for each 'T} E N, the restriction ¢ 11 := ¢1rxT~N is a Lagrange functional on T 11 N (see Definition 1.1.2)). Let ¢ = ¢(s, ~) be a Lagrange metric on M TM-+ R by

=I

x N. Define a map L :

(2.29)

where y = y 1 %. EB y 1 ~ E T(s,ry)(I x N). By Lemma 1.1.3, we know that Lis a singular Finsler metric on M. It is possible that L is singular in the directions y = ~ E T 11 N C T(s ,ry)(I x N). In many problems, however, the singularity does not cause much trouble. We can employ the methods developed in Finsler geometry to study Lagrange metrics. Example 2.1.8 In irreversible thermodynamics, the Riemannian approximation ignores the asymmetry of relativity entropy (information). If we take a consideration of the ignored items, say, the third term of the Taylor expansion, we obtain a Lagrange metric such as

¢ = Aab(s, 'TJ)~a~b

+ Aabc(s, 'TJ)~ae~c,

(2.30)

where a, b, c = 2, · · ·, n and det(Aab) > 0. This is studied by R. Ingarden and his collaborators (see [In] and references therein). If a "thermodynamic time" is introduced as a new coordinate, we obtain a Finsler metric in the form L = [Aab(x)yayb

yl

+ Aabc(x)yaybyc] 2 (y1)2

'

(2.31)

where a= 2, · · ·, n. Here (Aab(x)) is a symmetric matrix satisfying det(Aab(x)) > 0. If the second term is ignored, Aabc ~ 0, then L is approximately a Kropina metric. ti

CHAPTER 2. FINSLER SPACES

30

2.2

Spherical Metrics

In this section, we are going to introduce some important Finsler metrics on sn. By the maps '1/J± : Rn--+ S± in (2.13), we can express them in Rn. For y E TxRn = Rn , define

where I · I and ( , )denote the canonical Euclidean norm and inner product in Rn respectively. AE and BE are functions on TRn = Rn x Rn. For 0 < c: :S 1, define

+ JBJ0 + 2(1x14 + 2c:lxl2 + 1) AE(y)

vif=€2(x,y) lxl 4 + 2c: lxl 2 + 1 '

(2.32)

FE is a family of Finsler metrics on Rn . Note that L := (F1 ) 2 is just the Riemannian metric in (2.9). The family of Finsler metrics, FE, have the following important properties: (i) The geodesics of FE are straight lines in Rn as point sets; (ii) The flag curvature of FE is always equal to 1. We will give the notions of geodesics and flag curvature in later sections. To verify these facts directly, we might have to employ a computer program (such as Maple and Mathematica) on a fast computer. Define

+ JBJ0 2(1x1 4 + 2c:lxl 2 + 1). AE(y)

(2.33)

FE is reversible, i.e., FE(-y) = FE(y). The geodesics of FE are still straight lines in Rn (see Section 12.2). However, the flag curvature of FE is no longer constant. We only know partial information on the geometric properties of FE . The flag curvature of FE is negative near the origin, but always larger than -2. While the flag curvature approaches 1 near infinity. Let '1/J± : Rn --+ S~ c Rn+l be two diffeomorphisms defined in (2.13). Then '1/J+' '1/J- push the Finsler metric FE to a smooth Finsler metric on sn . The

resulting Finsler metrics on sn are projectively flat Finsler metrics of constant curvature 1. These important examples are constructed recently by R. Bryant [Br3) . Bryant's examples in dimension two are described in the following

2.2. SPHERICAL METRICS

31

Example 2.2.1 ([Br1)[Br2]) As we have mentioned above, in dimension two, the family of Finsler metrics in (2.32) are the pull-back of the Bryant metrics on S2 . Bryant defines his metrics in a different way. Let V be a three-dimensional real vector space and V ® C its complex vector space. Take a basis {e 1, e2, e 3} for V and define a quadratic Q on V x C by

(2.34) where u = uiei,v = viei and a.,/3 E R. For y E V \ {0}, let [y) := {ty , t > 0}. Then S := {[y), y E V \ {0}} is diffeomorphic to the standard unit sphere § 2 in the Euclidean space lR3 . For a vector v E V, denote by [y, v] E T[yJS the tangent vector to the curve c(t) := [y +tv] at t = 0. Note that [y, v] = [y', v'] if and only if y' = ay and v' = av + by for some a > 0 and b E R. Define F : TS -+ R by

F([y, v])

:=

n[

Q(y,y)Q(v,v)- Q(y,v) 2 _ iQ(y,v)] Q(y,y)2 Q(y,y) ,

(2.35)

where R[·) denotes the real part of a complex number. Clearly, F is well-defined. Assume that 1/31 :::; a. < ~. Then F is indeed a Finsler metric on S2 . Bryant has verified that F has constant curvature K = 1. ~

There are many other Finsler metrics of positive constant curvature. Recently, D. Bao and Z. Shen discover another family of Finsler metrics on S2 n+l with constant flag curvature K = 1. The metrics in three dimension are described in the following example. Example 2.2.2 (Bao-Shen[BaSh2]) Let ( 1 , ( 2 , ( 3 be the standard right invariant 1-forms on S 3 such that

is the standard Riemannian metric of constant curvature 1. For k

~

1, define

and Then (2.36) is a Finsler metric on S 3 . Bao-Shen show that Fk is of constant curvature K = 1 for any k ~ 1. Further, this family of Randers metrics are not locally projectively flat. Let (x , y, z , u ,v, w) be the standard coordinate system in TR3 = R 3 x R 3 . Let 'P± : R 3 -+ S~ denote the diffeomorphisms defined in (2.13). We can express

CHAPTER 2. FINSLER SPACES

32

F" on R 3 . Pulling back (i by

cp-; 1

onto R 3 , where

E

= ±, we obtain

-Edx - zdy + ydz x2 + y2 + z2 + 1 zdx - cdy - xdz x2 + y2 + z2 + 1 -ydx + xdy - cdz x2 + y2 + z2 + 1

(2.37) (2.38) (2.39)

Plugging them into a" and (3", we obtain Jk 2 (cu

+ zv- yw) 2 + k(zu- EV- xw) 2 + k(yu- xv + cw)2 1 + x2 + y2 + z2

-~

2.3

Eu+zv-yw 1 + x2 + y2 + z2 .

Hyperbolic Metrics

Let n be a bounded open domain with boundary an in Rn. Suppose that n is strictly convex, i.e., any line segment joining two points in n is strictly contained in n. Then for an arbitrary point p E n, there is an unique functional cp: Rn -7 (O,oo) such that

cp(>..y) = >..cp(y), and

cp(y-p)=1,

>.. > 0, y ERn

yEan.

n or an is said to be strongly convex if the above defined functional 'P is a Minkowski functional on Rn for some p E n. One can show that the strong convexity of n is independent of the choice of a particular point p E n. Let n be a strongly convex domain in a vector space R n. For 0 Rn, define F(y) > 0 by y x

+ F(y) = z E an.

f.

y E Tx n

~

(2.40)

2.3. HYPERBOLIC METRICS

33

We obtain a Finsler metrics F on n, which is called the Funk metric on D. Moreover, we have the following lemma due to Okada [Ok]. Lemma 2.3.1 (Okada) The Funk metric F on a strongly convex domain n C Rn satisfies (2.41)

Proof Since n is strongly convex, by assumption, there is a Minkowski functional cp on Rn and a point p E D such that

cp(y - p) = 1, Thus

y E 80.

cp( x + F~y) - p) =

(2.42)

1.

By differentiating (2.42) with respect to xj or yj, we obtain

(oj- p-zpxiYi ) 'Pz•(z ) = 0 (oj- p-I Fy;yi)'Pz'(z) = 0, where z := x

+ y j F(y) -

(2.43) (2.44)

p. It follows from (2.43) and (2.44) that

(Fxi - FFyi )'Pz' (z )yi Observe that

.

'Pz'(z)y'

= 0.

1

= cp(z)9z(z,y) =f. 0.

Thus This implies (2.41).

Q.E .D.

Let F be the Funk metric on a strongly convex 0

F(y)

F is called the

:=

c

R n. Define

~(F(-y) + F(y)).

(2.45)

Klein metric on 0 [Kl] . Using (2.41), we obtain

(2.46) The Funk metric F is positively complete with constant curvature K = -1/4 and the Klein metric F is complete with constant curvature K = -1. All the geodesics ofF and Fare straight lines. We will prove this fact using (2.41) and (2.46). See Section 4.3 and Theorem 12.2.11 below. See also [Fu1] [Bw3] [Bw4] [Bu3] [BuKe] [Ok] [Za] for related discussion.

CHAPTER 2. FINSLER SPACES

34

The reader can easily verify that the Funk metric F and the Klein metric on the unit ball Bn in lRn are given by

F

IYI2 - (1xl21yl2 - (x, y)2) + (x, y) F

1-lxl2 IYI2 _ (1xl21yl2 _ (x, y)2) 1-lxl2

(2.47)

(2.48)

where (,) and 1·1denote the standard inner product and Euclidean norm in lRn respectively. Note that F is just the Klein metric in (2.11) on Bn.

Chapter 3

SODEs and Variational Problems Second order ordinary differential equations (SODEs) arise in many areas of natural science. A special class of SODEs come from the variational problems of Lagrange metrics (including Finsler metrics). In general, it is impossible to find explicit solutions to a system of SODEs. Thus, one would like to know the behavior of the solutions based on some data of the system. In the early twentieth century, many geometers had made great effort to study SODEs using geometric methods. Among them are L. Berwald [Bwl, 1926; Bw7, 1947; Bw8, 1947], T.Y. Thomas [Thl, 1925; Th2, 1926; VeTh2, 1926], 0. Veblen [Vel, 1925; Ve2, 1929], J. Douglas [Dgl, 1928; Dg2, 1941], M. S. Knebelman [Kn, 1929], D. Kosambi [Ko2, 1933; Ko3, 1935], E. Cartan [Ca, 1933] and S.S. Chern [Ch2, 1939], etc. In this chapter, we will show that every system of SODEs (also called a semispray) can be studied via the associated system of homogeneous SODEs (also called a spray), and every Lagrange metric can be studied via the associated Finsler metric. Therefore, in the following chapters, we will be mainly concerned with sprays and Finsler metrics, rather than semisprays and Lagrange metrics.

3.1

SODEs

Let u = (a, b) X n, where system of SODEs on U

r

rf2 ds 2

n is an

open subset in Rn- 1 . Consider the following

df) = Pa ( s,f,ds '

a= 2,· ··,n,

(3.1)

where P : U x Rn - 1 --+ Rn- 1 are vector-valued functions. The graph of a solution J(s) = (P(s), .. . ,r(s)) is the curve c(s) := (s, J(s)) in u. Let S := (l,e,···,C,P 2 (s, ry,0,···,Pn(s,ry,~)). 35

36

CHAPTER 3. SODES AND VARIATIONAL PROBLEMS

S is a vector field on U x R n- 1 . Let ( s, 'I] a, ~a) denote the standard coordinate system in U x Rn- 1 . We can express Sin the following form

Sis called a semispmy on U. The notion of semisprays can be extended to fiber bundles 7f: M--+ (a, b). It is easy to see that j(s) is a solution of (3.1) if and only if its lift c(s) := ( 1, j(s), *(s)) is an integral curve of Sin U x Rn- 1 . We will show that every semispray or a system of SODEs can be converted to a spray or a system of homogeneous SODEs in a natural way.

R

A homogeneous system of SODEs on an open subset U C Rn is a system as follows (3.2) where Gi 's satisfy A> 0.

(3.3)

Because of the homogeneity of Gi, any solution x(t) = (x 1 (t), · · ·, xn(t)) of (3.2) has the following property: for any A > 0, x(t) := x(At) is again a solution of (3.2). Let G := (y 1 ,···,yn,G1 (x,y),···,Gn(x,y)). G is a vector field on U x Rn. Let (xi, yi) denote the standard coordinate system in U x Rn . We can express Gin the following form

G is called a spray on U . The notion of sprays can be extended to manifolds. See Chapter 4 below.

3.1. SODES

37

It is easy to see that x(t) is a solution of (3.2) if and only if its lift :i:(t) := ( x(t), ~~ (t)) is an integral curve of Gin U x Rn. Now we assume that two systems (3 .1) and (3.2) are related by (3.4)

Clearly, for a given set of functions a, there are infinitely many sets of homogeneous functions Gi satisfying (3.4). A special case is when Gi are given by 1 { G (x,y) = 0 ca(x, y) = -hlyla(s, 1], ~) where s

= x 1 ,1]a = xa and

~a=

yajy 1 .

Suppose that f(s) = (JZ(s), · · ·, r(s)) is a solution of (3.1) and s = s(t) is a solution of the following equation with ~: > 0 d2 s

dtZ + 2G

1(

2 df (s) ) ( ds s, f(s) , 1, ds dt ) = 0.

(3.5)

Let x 1 (t) := s(t) and xa(t) := r(s(t)) , a= 2, · · ·, n. It is easy to verify that x(t) = (x 1 (t), · · ·, xn(t)) is a solution of (3.2) with > 0. Conversely, suppose that x(t) = (x 1 (t), · · ·, xn(t)) is a solution of (3.2) with dit' > 0. First, we see that s = x 1 (t) satisfies (3.5) with ~; > 0. Note that s = x 1 (t) has the inverse t = t(s) . Then r(s) := xa(t(s)) satisfy

d:/

Thus f(s) = (f 2 (s), · · ·, r(s)) is a solution of (3.1). This proves (a) in the following lemma. The proof of (b) follows the same argument , so is omitted. Lemma 3.1.1 Let a = a(s, 1], ~) be a set of functions in (3.1} and Gi = Gi(x, y) a set of homogeneous functions in (3.2}.

(a) Suppose that a and Gi are related by (3.4}. Then for any solution x(t) = (x 1 (t) , ···,xn(t)) of (3.2} with dit' > 0, f(s) := (x 2 (t),···,xn(t)) is a

CHAPTER 3. SODES AND VARIATIONAL PROBLEMS

38

solution of (3.1),where s = x 1 (t) satisfies (3.5) with ~~ > 0. Conversely, for any solution f(s) = (j2(s), . .. ) r(s)) of (3.1) and any solution s = s(t) of (3.5) with~~ > o, x(t) := (s, P(s) , ... , r(s)) is a solution of (3.2) dx wzt· h lit> 0. 1

(b) Suppose that the solutions of (3.1) and (3.2) are related as in (a). Then illa and Qi are related by (3.4).

Therefore, we can study a system of SODEs (3.1) by investigating a system of homogeneous SODEs (3 .2) provided that illa and Gi are replaced by (3.4). Since there are infinitely many systems (3.2) satisfying (3.4) for a given system (3.1), we have to study the properties of (3.1) which are independent of the choice of a particular set of functions Gi satisfying (3.4). We will discuss this problem in Chapter 12.

Example 3.1.1 Consider a system of SODEs on an open subset U Rn. d2 a df ds2 -ill (s,f,dJ' a=2, · · · ,n,

nc

r -

= (a , b)

x

(3.6)

where

Define {

G 1 (x,y) = !Dbc(x)ybyc, ca(x, y) = Aa(x)ylyl

!(

+ Bg(x)ylyb + cgc(x)ybyc) .

(3.8)

Then illa and Gi satisfy (3.4). By Lemma 3.1.1, we can study (3.6) by investigating the following homogeneous system with regular coefficients d2x 1

{

di'2

d 2 xa ([t2-

dxb d xc

=0 Aa( X )dx' dx' Ba( )dx' dxb ca ( )dxb dxc _ o l i t l i t - b X l i t l i t - be X l i t l i t - .

+ Dbc

(

X

)

lit lit

(3 9)

·

For a single SODE on an open subset U = (a, b) x (c, d) C R 2 , we usually express it as follows d2 y ( dy) (3.10) dx 2 =ill x,y, dx ' where ill = ill(x, y, ~) is a following form ill= A(x, y)

coo

function on U x R. Assume that ill is in the

+ B(x, y)~ + C(x, y)e + D(x, y)e .

(3.11)

In this case, we can construct several homogeneous systems {

d2x (jj'I"

+ 2G ( x , y, dx dt) ~) dt

--

0

d2y (jj'I"

dx ~) + 2H( x, y, dt ' dt

-

0

-

(3.12)

3.1. SODES

39

such that G and H satisfy (3.4), namely, (x,y,~) = 2~G(x,y, 1,~)-

2H(x,y, 1,~).

(3.13)

Option 1:

{

G

= ~D(x, y)v 2

H

=

-HA(x, y)u

Option 2:

{

2

H

B(x, y)u 2

G

=

H

= -~A(x,y)u 2

+ B(x, y)uv + C(x, y)v 2 ) + C(x , y)uv + D(x, y)v 2

(3 .14)

(3.15)

Option 3: = iB(x,y)u 2 + iC(x , y)uv + ~D(x,y)v 2 = -~A(x,y)u 2 - iB(x,y)uv- iC(x,y)v 2

{G H

(3.16)

Notice that (3.13) is satisfied by any set of G and H in (3.14)-(3.16). Thus, by Lemma 3.1.1, we can study (3.6) by investigating either homogeneous system. When D = 0, we have more choices for regular systems whose coefficients G and H satisfy (3.13). U Some systems of first order ordinary differential equations can also be studied by converting them to systems of SODEs. Example 3.1.2 (Volterra-Hamilton System with Discounting [AnBu]) E. Baake and V. Kfivan independently suggest that a reasonable model for discounted production in ecology can be written as {

= k(a)Ya - h(a)Xa,

(not summed) ~ = 'Ytc(t, x)ybyc + 'Yg(t, x)yb + ea(t), dita

where a, b, c = 2, · · · , n and k(a) transformation

-=/::.

(3.17)

0, h(a) are the constants. Take the following (3.18)

The system (3 .17) becomes

{

~- ga

~: rgc(s, j)gbgc + rg(s, j)gb + ra(s, f),

where

k(~

rgc

k

rg ra

h(a)Jb

(b)

(c)

exp [(h(a)- h(b)- h(c))shtc(s, x)

+ kk(a) (b)

exp [(h(a)- h(b))sht(s, x)

k(a) exp(h(a)s). ea(s).

(3.19)

40

CHAPTER 3. SODES AND VARIATIONAL PROBLEMS

where xa = exp[-h(a)s]rya, a= 2, · · ·, n . Set ~a(s,ry,~) := rbc(s,ry)e~c

+ rb(s,ry)~b + ra(s,ry).

The system (3.19) becomes (3.20)

See [AnBr] for the systematic study on Volterra-Hamilton models in Ecology.

U

There are many systems of first order ordinary differential equations which give rise to a system of SODEs. For example, the Lorenz system

~~

{ 1f dz dt

= o-(y -

X)

= rx- y- xz - xy- bz -

(3.21)

is equivalent to the following SODE d2y

dx 2 dy +(x- 1 - ay)-

dx

+ 3y- 1 (dy)2 ,

(3.22)

dx

where j3 = -

a=

1

a2

, 'Y =

b(a

+ 1)

a2

'

is a Lagrange metric on U. Consider the following variational problem of¢> on U: [(f) =

Let

f : [a, b]

-t

nc

J

1> ( s, f (s),

ds (s)) ds = Extremum.

Rn- 1 be an arbitrary

H(u, s) off with

H(O, s) = f(s),

H(u,a)

coo

(3.24)

function. Take a variation of

= f(a),

H(u, b) = f(b).

The energy functional

E(u)

:=

rb ¢>(s,H(u,s),a-;(u,s))ds aH

la

satisfies

Let

V(s)

:=

8H

ou (0, s).

V(s) is a vector field along c(s) = (s,f(s)), a :S s :S b. We obtain (3.25)

Assume that f(s) is an extremal of (3.24). Then £'(0) = 0 for any variation

of

f. This gives rise to the following system of Euler-Lagrange equations c/>rya -

! (¢>~a)

= 0,

a = 2, ... 'n.

Let ( hab) := ( hab) - 1 . Expand (3.26) as follows

c/>rya - 1>st.a -

1>1/b~a

djb

ds -

d2 fb 2hab ds2 = 0,

a = 2, ... 'n.

(3.26)

CHAPTER 3. SODES AND VARIATIONAL PROBLEMS

42

We obtain (3.27) where (3 .28) Variational Problem II. Let U be an open domain in Rn. Consider a function L = L(x, y) on U x Rn satisfying (i) Lis C 00 on U x (Rn \ {0}), (iia) L is positively homogeneous of degree two, i.e.,

L(x, A.y) = >.. 2 L(x, y), (iib) for any y

f.

>..

> 0, (x, y)

E U x Rn.

0, they-Hessian 9ij(x,y) of Lis nondegenerate, where

By definition, L is a Finsler metric on U. Consider the following variational problem of L on U: L:(x) :=

J

L ( x(t),

~~ (t)) dt =Extremum.

(3.29)

Assume that x(t) is an extremal of (3.29). By a similar argument, one obtains the following system of Euler-Lagrange equations i = 1, · · · ,n.

(3.30)

Observe that

where (3.31) We obtain (3.32)

Lemma 3.2.1 For any solution x(t) of (3.30}, L(x(t),

~~ (t))

=constant.

(3.33)

3.3. RELATIONSHIP BETWEEN EULER-LAGRANGE EQUATIONS

43

Proof Expand (3.30) as follows

(3.34) Using (3.34), we obtain

Q.E.D.

3.3

Relationship Between Euler-Lagrange Equations

In the previous section, we studied the variational problems of a Lagrange metric ¢ = ¢(s , ry, ~) and a Finsler metric L = L(x , y) on an open subset U = (a, b) x n in Rn. We obtained the Euler-Lagrange equations for each of them, namely, we obtained (3.27) for ¢

and (3 .32) for L

where In this section, we will show that if ¢ and L are related by

L(x,y) :=

[y 1 ¢(s,ry,~)r,

(3.35)

where x = (s,ry) andy= y 1 (1 , ~). Then q>a and Gi satisfy (3.36)

44

CHAPTER 3. SODES AND VARIATIONAL PROBLEMS

Proposition 3.3.1 If a Lagrange metric¢ and a Finster metric L are related by (3.35), the corresponding functions q>a and Gi satisfy(3.36). Proof The system of Euler-Lagrange equations of¢ are given by

(3.37)

a= 2,· · ·,n.

The system of Euler-Lagrange equations are given by i = 1, · · · ,n.

(3.38)

By Lemma 3.1.1(b), it suffices to prove that the solutions of (3.37) and (3.38) are related as in Lemma 3 .1.1 (a) . 1 Assume that x(t) = (x 1 (t), · · · ,xn(t)) is a solution of (3 .38) with > 0. Lets= x 1 (t) . Note that ~: > 0. Thus the inverse function t = t(s) exists. Let f(s) = (x 2 (t(s)), · · · ,xn(t(s))). By Lemma 3.2.1,

dit

df (s) ) ] 2 =constant. L ( x(t), dx dt (t) ) = [ ds dt ¢ ( s, f(s), ds

For a generic solution x(t) , we may assume that L(x(t), ds ( df ) dt ¢ s, f(s) , ds (s) =constant

~~(t)) :f. 0. Then

:f. 0.

(3.39)

Let

F(x,y) := y 1 ¢(s,ry,~),

where x = (s,ry) andy= y 1 (1,0. (3.39) implies that

F( x(t), ~: (t)) =constant :f. 0. It follows from (3.38) that x( t) satisfies the following equation

i = 1, · · · ,n. Observe that

d~l [ cp a 17

1(¢€ a)]

=

~; cp

17

a-

! (¢€a)

= Fxa -

(3.40)

!(

Fya) = 0.

Thus f(s) satisfies (3.37) . Assume that f(s) = (j2(s), · · · , r(s)) is a solution of (3.37) and s = s(t) satisfies (3.39) with ~: > 0. Let x(t) := (s(t), P(s(t)), ... 'r(s(t))) . Observe that

3.3. RELATIONSHIP BETWEEN EULER-LAGRANGE EQUATIONS and

Fxa -

:t (

Fya) = [ cPrya -

! (cP€a)] ~:

45

= 0.

Thus x(t) satisfies (3.40). It must also satisfy (3.38) with

dJt > 0. 1

Q.E.D.

Example 3.3.1 Let¢= ¢(s,ry,~) be a Lagrange metric on U = (a,/3) x f! C Rn. By definition ,

det ( hab)

~

0,

where hab := ~cP€a€b (s, ry, ~). Let (3 = Aa(s, ry)~a be a family of 1-forms on f! and A= A(s , ry) a family of scalar functions on n. Define a map L: U x (Rn \ {y 1 = 0}) --t R by

where (s, ry) = (x 1 , · · ·, xn) and ~ = (y 2 jyl, · · ·, yn jy 1 ). Then L is a singular Finsler metric on U. In particular, if

where hab = hba and det(hab) ~ 0, then

where (s, ry) = (x 1 , · · ·, xn). Lis a Kropina metric on U.

Example 3.3.2 Consider the following Lagrange metric on an open subset U = (a, (3) x (>.. , Jl.) C R 2 :

¢ = a(x, y)e where a= a(x, y)

~

+ b(x, y)~ + c(x, y),

(3.42)

0. The Euler-Lagrange equation of¢ is given by

1 [ ( dy) 2 d2 y dx2 = - 2a ay dx

dy

]

+ 2ax dx + bx - Cy .

(3.43)

The Finsler metric L associated with ¢ is given by

[

v2

]2

L= a(x,y)-:;;+b(x,y)v+c(x,y)u .

(3.44)

Here we denote by (x, y, u, v) the standard coordinate system of TU = U x R2 . Observe that 2ac + b2 2 ) 12 ( 2 2 u , Lvv = u 2 a v + abuv + 6

46

CHAPTER 3. SODES AND VARIATIONAL PROBLEMS LuuLvv - LuvLuv

= 8a u 6 ( av 2+ buv + cu 2) 3.

Assume that b2

Then LuuLvv - LuvLuv

-

4ac

< 0.

(3.45)

> 0 for u f= 0, regardless the sign of a. Note that 2

(ab) 2 - 4a 2 Cac: b ) =

~ (b 2 -

4ac)

< 0.

Thus Lvv > 0 for u f= 0. This implies that if (3.45) is satisfied, L is a positive definite Finsler metric for u f= 0. Given arbitrary

coo

functions R, P and Q on an open interval I, let

b

J P(x)dx e ' Ey(x,y),

c

(2R(x)y- Q(x)y 2)ef P(x)dx

a

+ Ex(x , y),

where E(x,y) is an arbitrary coo function on an open subset I x (3.43) simplifies to d2y dy dx2 + P(x) dx + Q(x)y = R(x)

n C R2 .

Then (3.46)

and the corresponding Finsler metric L is given by

The Finsler metric L in (3.47) for (3.46) was constructed by M. Matsumoto (Mall]. According to the above argument, L is positive definite for u f= 0 on the domain n c R 2 if

Ey(x,y) 2 - 4ef P(x)dx [(2R(x)y- Q(x)y 2 )ef P(x)dx

+ Ex(x,y)] < 0.

We will see that this family of metrics are locally projectively flat. See Example 13.6.1. ~

Chapter 4

Spray Spaces In this chapter, we will introduce an important geometric structure on a manifold and discuss some basic properties. Roughly speaking, a spray on a manifold M is a family of compatible systems of 2nd order ordinary differential equations in local coordinates (4.1) where (ci(t)) denotes the coordinates of a curve c(t), and Gi(y) are positively homogeneous functions of degree two, i.e.,

>. > 0. The compatibility gives rise to a globally defined c= vector field G on the tangent bundle T M \ {0} which is expressed in local coordinates as follows

G

.a

.

a

= y'f'l"'2G'(y)f'l"'· ux' uy'

(4.2)

Conversely, given a globally defined coo vector field G on T M \ {0} in the form (4.2) with positively homogeneous coefficients Gi(y) of degree two, one obtains a family of compatible systems of 2nd order ordinary differential equations (4.1) in local coordinate system.

4.1

Sprays

First, let us define a spray precisely.

Definition 4.1.1 Let M be a manifold. A spray on M is a smooth vector field G on T M \ {0} expressed in a standard local coordinate system (xi, yi) in T M as follows G

. f)

.

f)

= y'f'l"'2G'(y)f'l"', ux' uy' 47

(4.3)

48

CHAPTER 4. SPRAY SPACES

where Gi (y) are local functions on T M satisfying

A.> 0.

(4.4)

A manifold with a spray is called a spray space. One is referred to (Lan] for some discussion on sprays.

If cis an integral curve of G, i.e.,

de

(4.5)

dt = G t,

then the projection c =

1r

o

c satisfies (4.6)

Conversely, if c satisfies (4.6), then its canonical lift of G.

c := c is an integral curve

Definition 4.1.2 A regular curve c in M is called a geodesic of G if it is the projection of an integral curve of G . We shall call Gi the spray coefficients of G. Example 4.1.1 Let V be ann-dimensional vector space. Let (xi, yi) denote a standard local coordinate system for TV : : : : V x V determined by a basis {ei}i= I for V. Then G - i a -Y

-a. x'

is a well-defined vector field on TV. The geodesics of G are straight lines in V. G is called the standard flat spray on V. Spray spaces can be viewed as U generalized vector spaces.

Example 4.1.2 Consider the following spray on R 3

G

a + v -a + w-a = u -ax ay az

2

2

a av

2(xu + zw )-.

A curve c(t) = (x(t),y(t),z(t)) is a geodesic of G if it satisfies

d2 x dt 2 d2y dt 2 d2 z dt 2

0,

(dx)2 - 2z (dz)2 dt '

- 2x dt 0.

49

4.1. SPRAYS

We obtain

c(t) =(a, b, c)+ (u, v, w)t + cp(t)(O, 1, 0),

where

~ ( u 3 + w 3 )t 3 -

cp(t) = -

(

au 2 + cw 2 ) t 2 .

Thus the geodesics of G are planar curves in R 3 .

Example 4.1.3 Consider the following spray on R2 {)

V;

{)

ox

where r

oy

r

> 0. The geodesic c(t) d2 x

dt 2

1

r

1

+-;;: (~f + (ftf~~ r

dt2

dx dt

U

-yu 2

(4.7)

=

o,

(~~r + (~~r ~~ = o.

= -A sin (~t +B) r

0 8v

+v 2 - ,

= (x(t), y(t)) of G satisfies

1 d2 y ---

We obtain

0

au +

G = u - +v-- -vu 2 +v 2 -

'

-t + () ) , -dy =A cos (A r dt

where A> 0.

dy . dx Integrating dt and dt y1elds x (t)

r cos ( ~ t + ()) - r cos() + x (0),

y(t)

rsin(~t+B)

-rsinB+y(O).

Thus the geodesics of G are circles of radius r.

(4.8)

50

CHAPTER 4. SPRAY SPACES

Below we are going to look at sprays from different points of view.

I. Horizontal Tangent Bundle Over the Slit Tangent Bundle. Let M be an n-dimensional manifold and T M \ {0} denote the slit tangent bundle. Take a standard local coordinate system (xi,yi) in TM. Let VT M :=span{

0~ 1 ,

· · · ,

a~n }·

(4.9)

VTM is a well-defined subbundle of T(TM \ {0}). We call VTM the vertical tangent bundle over T M \ {0}. There is a canonical vector field on T M \ {0} defined by ·0 Y ·(4.10) .- y" ayi.

Y is called the vertical radial field which is a section of VT M. Given a spray on M,

G = Let

. 8

8

.

y"~- 2G"(y)~.

ux"

uy"

. 8Gi NJ(y) := ~(y) . uyJ

(4.11)

We call Nj the connection coefficients of G. The homogeneity condition (4.4) implies (4.12) Nj(>.y) = >.Nj(y) , v>. > o, and

.

G"(y)

1

Let HTM :=span{

where

J ux"

.

.

= 2NJ(y)yJ.

8 ux"

J!l, ···, J!n }, .

8 uyJ

~ := ~- Nf(y)~.

(4.13) (4.14) (4.15)

'HTM is a well-defined subbundle ofT(TM\ {0}). We call 'HTM the horizontal tangent bundle over TM \ {0}. It follows from (4.13) that i J - y Jxi.

G-

(4.16)

Therefore G is a horizontal vector field which is a section of HT M . The above argument gives a direct decomposition of the tangent bundle of

TM\{0} . T(TM\ {0}) = VTM ffi 'HTM.

Y and G are the canonical sections of VT M and 'HT M respectively.

(4.17)

4.1. SPRAYS

51

Let

H*T M :=span{ dx 1 , · · ·, dxn },

(4.18)

V*T M :=span{ . > 0.

Define G=G-2PY.

Clearly, G is a spray on M. Moreover, G and G have the same geodesics as point sets. That is, every geodesic of G, after reparameterized, becomes a geodesic of G, and vice versa. In this case, G is said to be pointwise projectively related to G. More general, let Q = Qi (y) 8~, be a vertical vector field on T M \ { 0} which has the following properties (i) QisC 00 onTM\{O}, (ii) the coefficients Qi(y) are positively homogeneous of degree two, i.e.,

Define

G = G-2Q.

Again, G is a spray on M. In general, the geometric relationship between G and G is quite complicated.

II. Path Spaces. A spray G on a manifold M determines a collection of parameterized curves (called geodesics) in M. Conversely, a collection of parameterized curves with certain properties induces a spray. This will be explained below. Let 9 be a collection of C 00 parameterized curves a : (a, b) -+ M with the following properties

52

CHAPTER 4. SPRAY SPACES

(i) (Existence) For every vector y E TxM and any t 0 , there is a curve r : (a, b) -+Min 9 with t 0 E (a, b) and '}'(to) = y; (ii) (Uniqueness) For any two curves r(t),a < t 0. For simplicity, we will always denote by dx 1 · · · dxn the wedge product dx 1 1\ · · · 1\ dxn. A volume form on M is a collection 2 of coordinate volume forms {(U, df..l)} such that if U n U=j; 0, then df..l = u(x)dx 1 · · · dxn and djj, = a(x)dx 1 1\ . · · 1\ dxn are related by u

=

a-1 ~~I·

We will denote a volume form on M by its representative df..l on a coordinate neighborhood U . Every volume form df..L defines a measure f..l on M by f..LU) :=

L

fdf..l,

Definition 5.1.1 A spray m space is a triple (M, G, df..l), where G is a spray on M and df..l is a volume form on M . A Finsler m space is a triple (M, L, df..l), where Lis a Finsler metric on M and df..l is a volume form on M . There are two canonical volume forms on any positive definite Finsler space. Both of them reduce to the same volume form when the Finsler metric becomes

63

CHAPTER 5. S-CURVATURE

64

Riemannian. In what follows, we will discuss these two canonical volume forms induced by positive definite Finsler metrics. Let (M, F) be an n-dimensional positive definite Finsler space. Let {e;}i= 1 be an arbitrary basis for TxM and {wi}~ 1 the dual basis for T; M. The Busemann-H ausdorff volume form dpp is defined by

(5.1) where

Clp(x)

Wn

:=

Euclidean Vol(B~)

and

(5.2)

(5.3)

The Busemann-Hausdorff volume form dpp determines a measure J.lF which is called the Busemann-H ausdorff measure . H. Busemann proved that ifF is reversible, then J.lF is the Hausdorff measure of the induced metric dp. See [Bu1](Bu2]. When F is Riemannian , i.e.,

the Busemann-Hausdorff volume form reduces to the usual Riemannian volume form. More precisely,

(5.4)

Example 5.1.1 Consider a Randers metric F =a+ (3 on a manifold M with

llf311x :=

sup

y ETx M,a(y) = l

f3 (y) < 1.

Let dpp and dpa denote the Busemann-Hausdorff volume form of F and a respectively. Let {ei} be an orthonormal basis for (TxM, ax)· We may assume that f3x(Y) = llf311xY1 . Then B~ is a convex body in Rn given by

The Euclidean volume of B~ is given by n)

Vol ( Bx

=

Thus

Wn

(1 - llf311i )

.>;.±.!. •

n+ l

dpp = ( 1 -

llf311;) -

2

(5.5)

2

df.la·

(5.6)

5.1. VOLUMES

65

This implies (5.7)

Assume that M is compact. Then

and the equality holds if and only if (3 = 0.

Example 5.1.2 Let n be a strongly convex domain in Rn. The Funk metric F is defined by x

y

+ Fx(Y)

E an,

(5.8)

y E Txn:::::: Rn.

From the definition, the unit sphere Sx := Fx- 1 (1) of Fx at xis given by

Sx = an- {x}. Take the standard orthonormal basis {ei} f= 1 for R n. We have

hence Euclidean Vol(B~) =Euclidean Vol(n). The Busemann-Hausdorff volume form df..tF = u(x)dx 1

· · ·

dxn ofF is given by

Wn

u(x) = Vol(n) ·

(5.9)

This implies that the Busemann-Hausdorff volume of (n, F) is constant (5.10)

There is another canonical volume form on ann-dimensional positive definite Finsler space (M,F). Let {ei}f= 1 be an arbitrary basis for TxM and {wi}f= 1 the dual basis for M. For a non-zero vector y = yiei E TxM, put

r;

gij (y)

are

coo functions on R n \

{ 0}.

Let

(5.11)

66

CHAPTER 5. S-CURVATURE

Define (5.12) where Wn denotes the Euclidean volume of the standard unit ball Then-form

Iffin

in

IRn.

(5.13) is a well-defined volume form on M. jl,p comes from a special volume form on T M \ { 0}. To see this, take a look at the following natural direct sum of the

cotangent space T;(TM \ {0}) in a standard local coordinate system (xi,yi) in TM,

where

t5yi

:=

dyi

+ Nj(y)dxi.

Using this decomposition, define a Riemannian metric on T M \ {0} by

g := 9ii (y )dxi 0 dxi + 9ii (y )t5yi 0 t5yi.

(5.14)

We call g the Sasaki metric induced by F. The Riemannian volume of g at y E TM \ {0} is given by (5.15) We call dV9 the Sasaki volume induced by F. The Sasaki volume form can also be expressed in terms of the Hilbert form w := 9ij(y)yidxi (5.16) We should point out that P. Dazord discussed (dw)n in his Ph.D. thesis [Daz1]. He further defined the volume of a compact Finsler space (M, F) by Vol(M) := __!_ { Wn

isM

dV9 ,

(5.17)

where 1r: BM-+ M denote the unit ball bundle of M. Dazord actually defined the volume using the tangent sphere bundle. But his definition is essentially same as (5.17) due to the homogeneity of F. See [Daz2]. Observe that for a function f on M,

{

isM

1r*jdV9=wn { fdjl,p,

iM

where

d[Lp = iTp(x)dx 1 · · · dxn is just the volume form defined in (5.13). So with one step further from Dazord's definition, we obtain the special volume form d[lp on M.

5.1. VOLUMES

67

R.D. Holmes and A.C. Thompson [Tho] took a different approach to study Minkowski geometry and discovered this special volume form d{Lp. Consequently, d{Lp is called the Holmes- Thompson volume form in literatures. There might be many interesting volume forms associated with a Finsler metric F on a manifold M. Nevertheless, every volume form d11- on M is in the following form where

¢~"

is a positive

c= function .

Example 5.1.3 Let (V, F) be ann-dimensional Minkowski space. Fix a basis {ei}f= 1 for V which determines a global coordinate system (xi) in V. For any point x E V,

_ ( ) __ (O) _

Up X

-

-

!Jp

JBn

det (9ii(Y)) dy 1 · · · dyn

,

Wn

The volume of the unit ball B ofF with respect to d{Lp is given by Euclidean Vol(Bn) _ (B)_ -

/.1-F

Wn

i

Bn

d et (9ii y( ))d y1 · · · dy n .

Assume that F is reversible, i.e., F( -y) = F(y). According to [Du], the Santal6's inequality [MePa] implies

and the equality holds if and only if F is Euclidean. Thus

and the equality holds if and only if F is Euclidean. This implies that for any reversible positive definite Finsler metric F on a manifold M,

(5.18) and the equality holds if and only if F is Riemannian.

u

CHAPTER 5. S-CURVATURE

68

5.2

S-Curvature

Consider the trivial spray on Rn

a G --Y i -a. x' and a family of volume forms dJ.Lc: = CTc:(x)dx 1

· · ·

E

dxn on Rn, where

> 0.

(5.19)

df.Lc: determines the well-known Gaussian measures f.Lc: on Rn . Note that CTc:(x)-+ 0 as E -+ o+. For a fixed value of E, the decay rate of the Gaussian measure f.Lc: at x is defined by a(JE

yi

S x(Y) :=--(-)-a1. (x) = 2c(x, y), (JE

X

X

The decay rate Sx of J.Lc: approaches oo (in the radial direction) as x -+ oo. We can extend the notion of decay rate to spray m spaces. Let (M, G, dJ.L) be a spray m space. In a standard local coordinate system (xi,yi) in TM, put

G

.a

= y' -a x'.

.

a

- 2G' (y)a----,' y'

df.L

= CJ(x)dx

1

.. -dxn .

Define

.a

y = y' -a·lx E TxM.

x'

(5.20)

S has the following homogeneity property. S(>.y) = >.S(y),

>.>0, yETM\{0}.

(5.21)

The quantity S is originally defined by the author [Sh3] for Finsler spaces with the induced Busemann-Hausdorffmeasure. Sis called the m ean covariation in [Sh3] and the mean tangent curvature in [Sh5] . The mean tangent curvature is generalized to spray m spaces (M, G, dJ.L) for the first time in this book. We shall call S the S-curvature. The following four types of sprays m spaces are of interest: (S1) S(y) = Si(x)yi is a 1-form in y; (S2) S(y) = Si(x)yi is a close 1-form on M;

5.2. S-CURVATURE

(S3) S(y)

= Si(x)yi

69

is an exact form on M;

(S4) S = 0. By (5.20), we see that (S1) (S2) (S3) are independent of the underlying volume form dJ1. Clearly, if Gi(y) are quadratic in y E TxM, x E M, then S becomes a 1-form on M. Every spray on a regular measure space can be deformed to another spray with vanishing S-curvature. More precisely, we have Proposition 5.2.1 Let (M, G, df1) be a spray m space and

2S G:=G+--Y n+1 ' where Y = yi 8~, denotes the canonical vertical vector field on T M.

(5.22) The S-

curvature of (M, G, df1) always vanishes.

Proof In local coordinates,

(5.23) By (5.23), we obtain (5.24) The homogeneity of S implies

as .

8yiyt = S.

Thus

acm oym

acm oym

ym oa a oxm

acm oym

--=---S.

(5.25)

We obtain -

acm oym

ym oa a axm

S = - - - - - = - - - - - -S=O. With this, the proof is completed.

Q.E.D.

Example 5.2.1 Let F denote the Funk metric on a strongly convex domain in Rn. According to Section 4.3, the spray coefficients Gi ofF are given by

Gi=~Fyi 2 .

n

(5.26)

CHAPTER 5. S-CURVATURE

70 Differentiating (5.26) yields

8Gm = n+ 1 F. 2 8ym By Example 5.1.2, the Busemann-Hau sdorffvolume form dt-tF = up(x)dxi · · · dxn is given by up

=

Euclidean Vol(O) =constant.

Plugging them into (5.20), one obtains

S(y) -- n + 2 1 F(y) .

(5.27)

In this sense, the 8-curvature ofF is constant.

Example 5.2.2 Consider a Randers metric F = o:+ j3 on a manifold M, where o:(y) = Jaijyiyj is a Riemannian metric and j3(y) = biyi is a 1-form with

Define bi;j by

k 8bi bi;j := 8xJ -bk!'ij'

where !'Jk denote the Christoffel symbols of o:. Let

Let Gi and (;i denote the spray coefficients ofF and o: respectively (see (4.30)). They are related by (5.28) where

P : Qi:

2~ (rklYkYl- 2o:(y)brarpSplYl)

(5 .29)

o:air SrlYl ·

(5.30)

See [BaChSh1] for a proof. Differentiating (5 .28) with respect to yi, then taking summation over i, we obtain (5 .31) N:;: = J\T:;: + (n + 1)P. Let

5.2. S-CURVATURE

71

From Example 5.1.1, n+l

e7p(x) = ( 1-

IIJ311;(x)) -

2

e7a(x),

(5.32)

Note (5.33) Plugging (5.31) and (5.32) into (5.20) yields

S(y) = (n

1

2

.

]

+ 1) [ 2 (1 -II;3IIZ,(x)) (ll;311a)x,Y1 + P(y) ·

(5.34)

By an elementary argument, one can easily prove that the following conditions are equivalent for Randers metrics.

(i)

s=

0;

(ii) S is a 1-form; (iii) ;3 satisfies (5.35) Thus if

ll;3lla =constant,

(5.36)

u

then S = 0.

The condition (5.36) means that ;3 is a Killing 1-form of constant length. If ;3 is parallel, then it must be a Killing 1-form of constant length. There are some Riemannian spaces with non-parallel Killing 1-forms of constant length. See Example 5.2.3 below. Example 5.2.3 Let {( 1 , ( 2 , ( 3 } be the canonical left-invariant co-frame on the Lie group Sp(1) = S3 so that the following is the canonical Riemannian metric of constant curvature "' = 1 on S3 .

Consider F = o:

+ ;3,

o:(y) : ;3(y) :

where

Ja2[(l(y)J2 + b2[(2(y)J2 + c2[(3(y)J2, b1( 1(y) + b2( 2(y) + b3( 3(y) .

(5.38) (5.39)

CHAPTER 5. S-CURVATURE

72

Assume that

2)2 +(b-3)z +(blltJIIa = ( -~)z a b c

0. For a tangent vector y E Tx M, let Y be a geodesic field with Yx = y. Then (5.42) Proof Observe that in local coordinates

a~k ( ln Jdet(gij))

a~k ( ln Jdet(gij))

(5.43) (5.44)

where Cijk denote the coefficients of the Cartan torsion. It follows from (4.40) that ~ykgii 8gii _ 2gii C . Gl = Ni (5.45) ' . tJl 8xk 2 By definition, the geodesic field Y = yi 8~, satisfies (5.46)

74

CHAPTER 5. S-CURVATURE

Here Yin Gi stands for its coordinates (x 1 ,· ·· ,xn;Y 1 (x),···,Yn(x)). (5.43)-(5.46), we obtain

yk a~k (In

J

det(gii (Y))) - yk a~k (In (J)

~ yk ii (Y) [ 8giJ (Y) g

2

8xk

~ yk gii (Y) agii (Y) 8xk

2

By

l

+

.

2C. (Y) 8Y ] - yt 8CJ

tJl

8xk

CJ 8xi

- 2gii (Y)Ci ·1 (Y)G1(Y) - yi a(J. 1 CJ 8xt

Ni(Y)- yk a(J = S(Y). t CJ 8xk This completes the proof.

5.3

Q.E.D.

Geodesic Flows

In this section, we will discuss geodesic flows and the geometric meaning of the S-curvature. Let G be a spray on M . For any y E T M \ {0}, there is an unique curve c(t) = cy(t) in TM \ {0} satisfying

de dt

= Gt,

c(O) = y.

(5.47)

Let

-

n +- 1 p y'y'.U i V j . 2

(6.17) (6.18)

-

E = 0.

Therefore, G is an affine spray if and only if P is a 1-form on U. More general, consider two sprays G and G on a manifold M. Let

Q is a vertical vector field on T M \ { 0} . Take the vertical lift {ey} f= 1 of a basis {ei}f= 1 for TxM and express Q = Qi(y)ey . Qi are positively homogeneous of degree two, namely, Qi(>.y) = >. 2 Qi(y), V>. > 0. Let

. Q}kt(Y)

:=

83Qi 8 yi 8 yk 8 yz (y).

Define Qy(u,v,w) := Q~k 1 (y)uivkw 1 , where u = uiei, v = vie1 , w = wkek. Let B and B denote the Berwald curvature of G and G respectively. A direction computation yields (6.19) Consider a two-dimensional spray on an open subset U C R 2

8

8

8

8

G =u 8 x +v 8 Y -2G(x,y,u,v) 8 u -2H(x,y,u,v) 8 v.

Write G

= G(x, y, u, v)

and H

= H(x, y, u, v)

G

~e(x,y, ~) u

H

y ~) ~e(x ' 'u 2

in the following form

2

uv-

y ~) ~ 0 is a coo function.

u>O,

(6.50)

Note that

LuuLvv - LuvLuv = 4¢3 ¢£.£.·

Thus (6.44) is equivalent to that

G

¢c.c. > 0.

From (6.45) and (6.46), we obtain

= ~e(x,y,~)u 2 ~e(x y ~) uv- ~1>(x y ~) 2 ' 'u 2 ' 'u

H

u2

'

where

1>

=

e =

c/Jy- c/Jxe, - c/Jyt,~ ¢c.c. q> c/Je, + ¢x + c/Jy~ .

¢

(6.51) (6.52)

¢

Differentiating G and H gives

Plugging them into (6.48) and (6.49) yields

4~ [ew + (3ee,e,- 1>w)

J(y)

-

E(y)

4~ [30e,e, - 1>m] ¢~c. .

;J (¢~c.) ~ ,

(6.53) (6.54)

Thus J = 0 if and only if (6.55)

CHAPTER 6. NON-RIEMANNIAN QUANTITIES

92 and E = 0 if and only if

38€€ - denotes the Levi-Civita connection of geodesic with respect to g.

7.2

g

:= gy.

(7.15) Hence Y is also a

Chern Connection of Finsler Metrics

In the previous section, we introduced the Berwald connection for sprays. Since every Finsler metric induces a spray, the Berwald connection is also defined for Finsler metrics. In 1934, E. Cartan became interested in Finsler spaces [Ca]. It was the fashion then that connections be metric-compatible such as the Levi-Civita connection in Riemannian geometry. So Cartan modified the Berwald connection for Finsler spaces and introduced the Cartan connection. The Cartan connection is indeed metric-compatible, but not torsion-free. Moreover it can be defined only on an appropriate vector bundle (e.g. the vertical tangent bundle) over TM \ {0}. In 1943, S. S. Chern found another notable connection for Finsler metrics [Ch1][Ch3] . The Chern connection is different from the Berwald connection only by a quantity - the Landsberg curvature. The Chern connection is also called the Rund connection in literatures, because Chern's papers were written using Cartan's exterior differential methods which were not known to other geometers in Finsler geometry. The curvatures of both the Chern connection and the Berwald connection are in very simple form, since they are all torsionfree. In this section, we will give a brief description of the Chern connection. See [BaCh][BaChSh1] for more details. Let (M, L) be a Finsler space. For a vector y E TxM \ {0}, the Landsberg curvature Ly : TxM x TxM x TxM -t R determines a bilinear symmetric form

CHAPTER 7. CONNECTIONS

100

gy(Ly(u, v), w)

= Ly(u, v, w) .

(6.28) implies

Ly(y, v) Define a map fjY: TxM

X

= 0.

(7.16)

C 00 (TM)-+ TxM by (7.17)

'lltV := VtV- Ly(u, v),

where u,v E TxM and V E C 00 (TM) with Vx = v. We call 'IJ = the Chern connection. In a standard local coordinate system (xi, yi) in T M, let

{'IJY}yETM\{0}

where L)k := gil Ljkl· We can express the Chern connection as follows

V

/ /

v

= Vx

u

It follows from (7.16) and (7.17) that

(7.18) (7.19) where Y, V E C 00 (TM) and w,y = Yx E TxM. The Chern connection is characterized by the following equations -y

-y

'V u V- 'V vU

-y w[gy(U, V)] = gy('V wU, V)

= [U, V],

-y + gy(U, 'V wV)

-y + Cy(U, V, 'V wY),

(7.20) (7.21)

where wE TxM and U, V, Y E C 00 (TM). Clearly, (7.21) is simpler than (7.11).

7.3. COVARIANT DERIVATIVES ALONG GEODESICS

101

The Chern connection also gives rise to a new quantity i ·at~k i Pjkl(Y) . - - ayl (y)- -Bjkl(Y)

Note that

i

l_

PJklY - 0,

j_

i

+

aL~k

(7.22)

ayl (y).

i

PJklY - -Lkl·

For a vector y E TxM \ {0} , define Py: TxM ® TxM ® TxM-+ TxM by

Py(u, v, w) We call P =

{Py}yETM\{O}

a = PJk· 1(y)u 1·v k w1 axi ix·

(7.23)

the Chern curvature. One can easily verify that

B= 0

{:=:?

P = 0.

Thus the Berwald metrics can also be characterized by P = 0. Connections are just tools. Since the Berwald connection is defined not only for Finsler metrics, but also for sprays, we will use the Berwald connection throughout this book.

7.3

Covariant Derivatives Along Geodesics

Once given a connection on a manifold, one can define the covariant derivatives of tensor fields along a curve. Let G be a spray on ann-manifold and f)k the Christoffel symbols of Gin a standard local coordinates in T M. Let c : [a, b] -+ M be a coo curve (possibly c(t) = 0 for some t). A vector field V = V(t) along cis a family of vectors V (t) where Vi(t) are

coo.

aI = V'.(t) !l"'" ux•

c(t)

,

Hence the tangent vector field of c,

is a vector field along c. Fix a vector field V = V(t) =1- 0 along c. For a vector field U(t) along c, we define

vru(t) := { ddUi (t) t

+ Ui(t)qk(V(t)) ddck (t)} !'>a I . t

ux'

c(t)

(7.24)

At a point c(to) where c(to) = 0, (7.24) reduces to

v

dUi.

aI

V,; U(to) = -d (to)!l"'" t ux'

c(to)

.

(7.25)

CHAPTER 7. CONNECTIONS

102 For simplicity, we always denote

DtU(t)

:= \7~U(t).

(7.26)

In local coordinates,

dU; DtU(t) = { dt(t)

.

.

+ U3 (t)NJ(c(t))

}8

(7.27)

fJxi lc(t)·

DcU(t) is called the covariant derivative of U(t) along c. Assume that cis a regular curve. Using 2Gi = Njyi, we obtain (7.28) Thus c is a geodesic if and only if

Dec=

o.

(7.29)

Equation (7.29) is just (4.6) in an index-free form.

Definition 7.3.1 A vector field V

= V(t)

along cis said to be parallel if

DtV = 0. Let p = c(a) and q = c(b). We define a map Pc : TpM-+ TqM by

Pc(v)

:=

V(b),

where V(t) is a parallel vector field along c with V(a) linear map . We call Pc the parallel translation along c.

= v.

Clearly, Pc is a

Lemma 7.3.2 Let (M , L) be a Finsler space. Assume that c : [a , b] -+ M is

a geodesic from p to q. Then the parallel translation Pc preserves the inner products 9c, i.e., 9c(b) (P(u), P(v))

= 9c(a)(u, v),

(7.30)

Proof Let U = U(t), V = V(t) be two parallel vector fields along c. By (7.11), we have :t (gc(U,V))

= 9c(t) ( DtU, V) + 9c(t) ( U, D,Y) = 0.

Thus 9c(U, V) is a constant. This implies (7.30)

Q.E.D .

There is another way to define the covariant derivative of a vector field U(t) along a curve c. (7.31) DtU(t) := \7fU(t).

7.3. COVARIANT DERIVATIVES ALONG GEODESICS

103

In local coordinates, _ { dUi D.:U(t) = -d (t)

t

. . }a I + d(t)Nj(U(t)) -a . . x• c(t)

(7.32)

A vector field V = V (t) f=. 0 along c is said to be auto-parallel if

D.: V(t) Define a map

Fe: TpM-+ TqM

= 0.

by

Fe(v ) = V(b), where V = V(t) is a auto-parallel vector field along c with V(a) = v. It is easy to see that Fe : TpM \ {0} -+ TqM \ {0} is a diffeomorphism satisfying

Fe is not linear in general. We call Fe the auto-parallel translation along c. Lemma 7.3.3 Let (M , L) be a Finster space and c : [a, b] -+ M a geodesic from p to q. Then the auto-parallel translation Fe : TpM-+ TqM preserves the Minkowski functionals. Proof It follows from (7.11) that for any auto-parallel vector field V along c,

!

Thus F 2 (V)

= gv(V, V)

(gv(V,

= V(t)

v)) = 2gv(D.:V, v) = o.

is a constant. This proves the lemma.

Q.E.D.

By Lemmas 7.3.2 and 7.3.3, the reader can easily see that for Berwald metrics, Pe = Fe is a linear transformation preserving the Minkowski functionals Fx in Tx M. This implies the following Proposition 7.3.4 ([Icl]) On a positive definite Berwald space, Minkowski tangent spaces are linearly isometric to each other by parallel translations along geodesics.

Roughly speaking, on a positive definite Berwald space (M , F), the geometry of Minkowski tangent spaces does not change. Thus S = 0. More precisely, we have Proposition 7.3.5 ([Sh3]) On any positive definite Berwald space (M,F), the S-curvature S = 0 for the Busemann-Hausdorff volume form df..Lp and the Holmes-Thompson volume form dji,p . Proof Let 0 f=. y E TxM be an arbitrary vector. Extend it to a geodesic field

Y in a neighborhood of x. Let c be the geodesic with c(O) = y and Pt := Pe,

denote the parallel translation along Ct := cl[o,t]. Take an arbitrary basis {ei}r=l for TxM. We obtain a parallel frame along c, i

= 1, · · · ,n.

CHAPTER 7. CONNECTIONS

104

e 1__________________ e2 _ -------------

-

X=

y

e1 (t)

---------eiUJ

---------------

c(O)

Let J.tF denote the Busemann-Hausdorff measure ofF which is defined in (5.1)(5.3). Put d~-tFic(t) = ap(t)w 1 (t) 1\ · · · 1\ wn(t), where {wi(t)} be the basis for Tc(t)M which are dual to {ei(t)}f= 1 .

According to Lemma 5.2.2, the S-curvature ofF in the direction of c(t) is given by S(c(t)) =

! [In (

det [gc(t)(ei(t), ej(t))] )] ap(t) ·

By Lemmas 7.3.2 and 7.3.3, we know that

Thus det [gc(t)(ei(t),ej(t))] = det [gy(ei,ej)],

B~(tl

=

{cvi), F(viei(t)) < 1} = {cvi),

F(viei)
ux•

i

j

k

l

J

Rj kldx 0 dx 0 dx 0 Jxi "

It follows from (8.17) that

(8.18) (8.19) The local quantities R~ (y), R\1(y) and R/ kl (y) all live on T M \ {0}. They are actually the coefficients of tensors on T M \ { 0}. See Chapter 9 below. A direct computation using (8.17) yields (8.20) Thus

R i kl = R j i klY j ,

Rik_ R j i kl yiyl .

(8.21)

111

8.1. RIEMANN CURVATURE OF SPRAYS

Remark 8.1.3 Every spray determines a unique horizontal distribution HT M span{ 8 ~,} on TM \ {0} (see (4.14)) . Observe that

J ) [ J Jxi' Jxk =

(JNJ

JN~)

Jxk - JxJ

a

i

=

a

ayi = -R kl ayi ·

Thus R = 0 if and only if HT M is integrable, namely, at every point y E T M \ { 0}, there is an n-dimensional su bmanifold N of TM passing through y such that the tangent space of Nat every pointy' EN coincides with Hy'™· This observation was made by X. Mo for Finsler metrics (Mo]. Assume that a spray G is affine. Then the Berwald connection \7 of G is affine. The Berwald connection coincides with the N-connection D. Since the Christoffel symbols qk are functions of x only, (8.20) simplifies to

. R/kl(x)

=

aql aqk axk (x)- oxl

.

.

+ rA,.(x)r_jl(x)- r_jk(x)fls(x).

(8.22)

Note that R/kl(x) live on M. We obtain a tensor Ron M

R(u, v)w

:=

a ix, R/· k1(x)u k v 1wl oxi

(8.23)

where u = uk~lx,v = v 1 ~1 x and w = wi 8~, lx· We call R the Riemann curvature tensor of D. The Riemann curvature tensor can be expressed in an index-free form. R(U, V)W

where U

= (DuDv- DvDu- D[U,vJ)w,

= Uk~, V = V 1 /xr

and W

= Wi 8~,

are vector fields on M,

Now we derive a useful formula for the Riemann curvature. Let G = yi 8~,2Gi(y) 8~, be a spray on a manifold M and Q = Qi(y) 8~, a vertical vector field on TM \ {0} with

.\ > 0.

Let We obtain a natural co-frame {wi,wn+i}j= 1 on TM \ {0}. Let W j i ·. --

r ijk dxk .

For any vertical vector field X = Xi 8~, on T M , the following map from tangent vectors to vertical tangent vectors on T M \ {0}

\lX

:= {

dXi

+ XJw/}

0

()~i

(8.24)

CHAPTER 8. RIEMANN CURVATURE

112

is independent of standard local coordinate system (xi, yi) in T M. We call \7 the Berwald connection and the Berwald connection forms. Define Q\, Qik, Q\.1 and Qik·l by

w/

'

'

'

(8.25) dQi ·k

+ Qj·k W·iJ

_ Qi.w j = Qi. wl ·] k ·k,l

+ Qi·k·l wn+l.

(8.26)

Q\ and Qtk, etc. are called the coefficients of the covariant derivatives of Q = Qi 8~, with respect to the natural basis { 8~, } for VT M . See Section 9.1 for more details . Based on (8.14), we introduce a new quantity

.a

u = u•8 x'-lx E TxM, where Hk := 2Q:k- Q\,iyi

(8.27)

+ 2QiQii·k- QiiQik.

(8.28)

By (8.14), we immediately obtain the following Lemma 8.1.4 The Riemann curvature of G := G- 2Q is related to that of G by (8.29) R=R+H. Take an arbitrary local frame {e;} for TM. Let {ei} denote the vertical lift of {e;} for VTM . If e; =a{ 8~.1x, then ei :=a{ 88 , ly, where y E TxM \ {0} . By the canonical bundle isomorphism HT M --+ M, we obtain a horizontal frame {ef} for HTM. If e; := ai 8~, lx, then ef :=a{ 0 ~, ly, where y E TxM\ {0}. Let {wi,wn+i} denote the co-frame dual to {ef,ei}. The Berwald connection \7 determines a set of connection forms w j i by

VT

Express Q = Qiei. One can define the covariant derivatives with respect to {wi,wn+i} using w{ Hy(u) = Hk(y)uke; can be expressed in terms of Qtk and Q\ etc. by the same formula (8.28), where Qtk and Q\, etc. are the coefficients of the covariant derivatives of Q = Qiei with respect to {ei}. Of course, (8.29) still holds. The formula (8.29) is useful in computing the Riemann curvature of certain Finsler metrics, such as Randers metrics. Definition 8.1.5 A spray on a manifold M is said to be isotropic if the Riemann curvature is in the following form Ry(u) = R(y) u

where Ty E

r; M

with Ty(y)

+ Ty(u)

= -R(y).

(8 .30)

y,

It is said to be R-fiat if R

= 0.

8.1. RIEMANN CURVATURE OF SPRAYS

113

A spray is said to be fiat if at every point, there is a local coordinate system in which i 8 G --Y~ · vx'

Clearly, flat sprays must be R-flat. The converse is not true. Can you find a R-flat spray which is not flat? The following proposition is due to L. Berwald and J. Douglas. Proposition 8.1.6 A spray is fiat if and only if B = 0 and R = 0. Proof We shall only sketch the proof here. First, B = 0 implies that the Berwald connection \7 is affine. In this case, the Berwald connection coincides with the N-connection D. From the definition of 1 , R = 0 implies that the Riemann curvature tensor R of D vanishes. A classical theorem on linear connections implies that there is a local coordinate system in which = 0. Thus G is flat. See [BaChShl] for details. Q.E.D.

R/k

qk

Definition 8.1.7 The trace of the Riemann curvature Ric(y) := (n- l)R(y) =

(8.31)

R~(y)

is called the Ricci curvature and R(y) = n~l Ric(y) is called the Ricci-scalar. Example 8.1.1 Consider the following spray on R 2

where

-. 2 ~.p(x,y,u,v),

{0}) satisfying

\

>-.

> 0.

A direct computation yields Ric=

2~.py(x,

y, u, v).

Thus Ric = 0 if and only if I{' is independent of y. Definition 8.1.8 A spray G is said to be Ricci-constant if the Ricci scalar R satisfies 0• R ;i ·(8.32) . - R xt· - NiR i y 1· It is said to be weakly Ricci-constant if (8.33)

It is said to be Ricci-fiat if R = 0.

CHAPTER 8. RIEMANN CURVATURE

114

Lemma 8.1.9 Suppose that a spray G is weakly Ricci-constant. Then for any geodesic c, Ric (c(t)) =constant.

Proof Observe that

!

ciRx; (c(t))

[R(c(t))]

+ ciRy; (c(t))

ciRx; (c(t))- 2Gi(c(t))Ryi (c(t))

= o. Q.E.D .

This proves the lemma.

Note that R = 0 implies that Ric = 0. The converse is not true, even in dimension two. See Example 8.1.2 below.

Example 8.1.2 Consider the following affine spray on R 3 [)

G = uox

[)

[)

oy

[) z

+ v - + w-

28

2[)

28

- 2au - - 2bv - - 2cw - , ow ov au

(8.34)

where a= a(x,y,z), b = b(x,y, z) and c = c(x,y, z). A direct computation yields R~ = 2az u 2 R~ = 2ay u 2 , R~ = -2ay uv- 2az uw,

= 2bx v 2 , = 2cx w 2 ,

R~

Ri Rr

R~

= - 2bx uv - 2bz vw , R~ = 2bz v 2 = 2cy w 2 , R~ = -2cx uw- 2cy vw.

Thus R = 0 if and only if

a= a(x),

b = b(y),

c

= c(z) .

The Ricci curvature is given by

Thus Ric = 0 if and only if

There are a = a(x, y, z), b = b(x, y, z) and c = c(x, y, z ) such that the corre~ sponding spray G is Ricci-fiat, but not R-fiat.

By definition, the Ricci curvature is the trace of the Riemann curvature. For a two-dimensional spray, the Riemann curvature can be expressed in a special form.

8.1. RIEMANN CURVATURE OF SPRAYS

115

Lemma 8.1.10 On a surface M, the Riemann curvature of a spray must be in the following form (8.35) Ry = R(y) I+ Ty y,

where R = Ric denotes the Ricci scalar, I : TxM --* TxM denotes the identity map and Ty E r; M is a linear functional satisfying Ty(y) = -R(y).

(8.36)

Proof To prove (8.35), we just need the fact Ry(y) = 0. Take an arbitrary basis { e1 , e 2 } for TxM and let y = ue 1 + ve 2 . Observe that R = R} + R~ and Riu - (R~ - R)v R~v- (R~ - R)u

= Riu + R~v = = R~u + R~v

0,

= 0.

Let

=

T1 : T2:

Ri v R~ u

Rl-R u R~ -R v

(8.37) (8.38)

It follows from (8.37) and (8.38) that for a vector u = rye 1 + (e2 E TxM,

R~17

+ R~(

Rry+ (T11J+T2()u,

(8.39)

Ri11

+ R~(

R(

+ (T11J + T2()v.

(8.40)

These two identities are just (8.35).

Q.E.D.

Now we compute the Riemann curvature of a spray G on a surface M. In a standard local coordinate system (x,y,u,v) in TM, express G by

a+ a v oy -

G = u ox

a

a

2G(x, y, u, v) ou - 2H(x, y, u, v) ov.

(8.41)

Using (8.14), we obtain

+ 2GGuu + 2HGuv- GuGu- GvHu -(Gxv- Gyu)U + 2GGuv + 2HGvv- GuGv- GvHv -(Hyu- Hxv)V + 2GHuu + 2HHuv- HuGu- HuHv (Hyu- Hxv)U + 2GHuv + 2HHvv- HuGv- HvHv.

R~

(Gxv- Gyu)V

R~ Ri R~

The Ricci curvature Ric = R = R}

R

=

+ R~

is given by

2Gx + 2Hy- G~- H~- 2HuGv- u(Gu + Hv)x -v(Gu

+ Hv)y + 2G(Gu + Hv)u + 2H(Gu + Hv)v·

(8.42)

CHAPTER 8. RIEMANN CURVATURE

116 Let

v 2 P := -G- (Gu + Hv) = -Gv - Hv. u u R can be expressed by

(8.43)

2 v R = -2-Gy + 2Hy + Pxu + Pyv- 2GPu- 2HPv- P . u

If P = 0, then

v R = -2-Gy u

+ 2Hy .

(8.44)

(8.45)

Now we express the spray coefficients G and H in the following form G

~8(x,y, ~)u 2

H

~~(x ~8(x u 2, ,y ,~)u 2 u ,y ,~)uv2

where 8 = 8(x,y,0 and~= P in (8.43) simplifies to

~(x ,y, ~)

P =

are functions of (x,y,(). The function

~( ~€- 8 ) .

It follows from (8.44) that R in the direction (u , v) = (1, ~) is given by

R =

-~y

1

+2~(~{-

When P = 0, i.e., 8 = is given by

~€,

1

1

+ 2(~€- 8)y~ + 2(~€ 8){-

1

4(~{

- 8)x

+ 8)(~{- 8).

(8.46)

the Riemann curvature in the direction (u, v) = (1, ~)

R~

~( ~x{{ - 2~y{ + ~Y€€~)~ + ~~~W~,

R~

-~ ( ~x€{- 2~y€ + ~Y€€~)~- ~~~€€€~- ~y·

R~ and Rr are given by R~ = -RU~ and Rr = -R~~ - The Ricci curvature in the direction of (u, v) = (1, 0 is given by

(8.47)

By the above formulas , we immediately obtain the following Proposition 8 .1.11 Let G = u %x + v spray, where G and H are in the f orm

G H

:Y-

2G %u- 2H %v be a two-dimensional

v) v) u , 1 (x, y,-;;, v) uv- 2~ (x, y,-;;, 1 2~~

2 ( 1 2 ~~ x,y,-;;, u

2

(8.48) (8.49)

8.2. RIEMANN CURVATURE OF FINSLER METRICS

117

where cl> = .P(x, y , ~) is an arbitrary function . The Riemann curvature vanishes if and only if .Py = 0, (8.50)

Let k

of. 0 be an arbitrary constant and

f(s) a solution to the following ODE

f(s)f"'(s) = kf"(s). Let

(8.51)

1

(8.52) kx It is easy to verify that cl> satisfies (8.50). Thus the corresponding spray G defined in (8.48) and (8.49) has vanishing Riemann curvature. This spray is not projectively affine, because no solution of (8.51) is a polynomial of degree three or less. See Corollary 13.2.5. q> : = - f(~).

Example 8.1.3 Let IB\ 2 be the standard unit ball in ll~?. Let

·- Ju 2 r.p (x,y,u , v ) .-

+ v2 -

(xv- yu) 2 +(xu+ yv) . 1- X 2 - y 2

r.p is a function on TIB\ 2 = IB\ 2 x R2 . Let G be a two-dimensional spray on IB\ 2 , a ax

G = u-

a

a

a + v - - 2r.pu-- 2r.pv-. oy au av

It is easy to verify that R = 0.

8.2

Riemann Curvature of Finsler Metrics

Every Finsler metric induces a spray by (4.29) and (4.30). Thus the Riemann curvature of the Finsler metric is defined to be that of the induced spray. Let us take a look at the following example before we discuss some special properties of the Riemann curvature for Finsler metrics. Example 8.2.1 Consider a Randers metric F = a+ (3 on an n-dimensional manifold M, where a= Jaij(x)yiyi is a Riemannian metric and (3 = bi(x)yi is a 1-form satisfying

llf311x

:=

Jaii(x)bi(x)bj(x) < 1.

It is natural to express the Riemann curvature of F in terms of that of a and the covariant derivatives of (3. In general, it is very complicated. A complete solution is given in [YaSh][ShKi][Ma6], etc. When (3 is a close 1-form, the formula for the Riemann curvature is much simpler.

118

CHAPTER 8. RIEMANN CURVATURE

Assume that j3 = bi(x)dxi is close. Let bi;j and bi;j;k denote the components of the covariant derivatives of j3 with respect to a. Put (8.53) Let Gi and (;i denote the spray coefficients of F and a respectively. According to Example 5.2.2,

. = Gt-.+ -1>yt.

Gt

2F

.

Plugging them into (8.14) yields

_. + [3(1>)2 w].hk Rk. = Rk -4 -F - -2F '

(8.54)

where In a index-free form,

Ry = Ry +

[~

(;f- 2~

](1- F- 2 (y)gy(y, ·)y).

(8.55)

It follows from (8.55) that

Ric

= Ric + (n

- 1) [ ~ (;)

2

-

2~] .

(8.56)

Without assuming that j3 be close, the formulas for R and Ric are much more complicated. See [YaSh][Ma6][Ma8] . We can compute them using (5.28) and (8.29). A good example is given in [BaSh2] where we compute the Riemann curvature for a special family of Randers metric on S3 . An important fact on Finsler sprays is that the Riemann curvature Ry is self-adjoint with respect to 9y · More precisely, we have the following Proposition 8.2.1 Let (M,L) be a Finsler space. For any y E T xM \ {0}, satisfies (8.57) gy(Ry(u),v) = gy(u,Ry(v)).

To prove (8.57), one just needs to verify the following gilRi

= 9klRi.

(8.58)

The detailed argument is given in Remark 8.4.4 and Section 10.1 below. From Proposition 8.2.1 , we see that the Riemann curvature Ry : TxM -+ T xM is self-adjoint with respect to gy for every y E TxM \ {0}. This implies that Ry is diagonalizable as a linear transformation. There are sprays whose Riemann curvature Ry is not diagonalizable. See Example 8.2.4 below. Thus not all sprays can be induced by a positive definite Finsler metric. Now let us take a look at Finsler metrics whose sprays are isotropic.

8.2. RIEMANN CURVATURE OF FINSLER METRICS

119

Lemma 8.2.2 Let L be a Finsler metric on a manifold M. Suppose that the

induced spray is isotropic, then there exists a scalar function >..(y) on TM \ {0} such that Riemann curvature is in the form Ry(u)

= >..(y){ gy(y,y)u- gy(y,u)y },

(8.59)

Proof By assumption, the Riemann curvature is in the following form Ry = R(y) I+ Ty y,

y E TxM \ {0},

(8.60)

where Ty E r;M with Ty(y) = -R(y). It follows from (8.60) that for arbitrary u,v E TxM,

R(y)gy(u, v) R(y)gy(v, u)

gy(Ry(u) , v) 9y (Ry (v), u)

+ Ty(u)gy(y, v), + Ty(v)gy(y, u).

By Proposition 8.2.1 , we obtain

Ty (U )gy (y, V)

= Ty (V )gy (y, u).

Taking v = y yields

Ty(u) =

R(y) Ty(y) ) 9y(y, u) = - L( ) 9y(y, u). ( y 9y y, y

Letting K(y) := R(y)/L(y), we obtain (8.59).

Q.E.D.

Lemma 8.2.2 leads to the following

Definition 8.2.3 ([Bw3][Bw7]) A Finsler metric L is said to be of scalar curvature if there is a scalar function K (y) on T M \ {0} such that the Riemann curvature is in the following form

Ry(u) = >..(y){ gy(y,y)u- gy(y,u)y },

(8.61)

The function >..(y) is called the curvature scalar . Lis of constant curvature>.., if >..(y) = >... Lis of zero curvature if>..= 0. Let L be a Finsler metric on a surface M. By Lemma 8.1.10, we know that the Riemann curvature of L is in the form (8.35). By Lemma 8.2.2, we conclude that Lis of scalar curvature with curvature scalar K(y) and the Riemann curvature is given by

Ry(u)

=

K(y){ L(y) u- gy(Y, u) y }·

(8.62)

We come to the same conclusion that every two-dimensional Finsler metric is of scalar curvature. Note that the Ricci curvature Ric(y) is related to K(y) by

K( ) = Ric(y) L(y) . y

(8.63)

CHAPTER 8. RIEMANN CURVATURE

120

The scalar function K(y) on TM \ {0} is the Gauss curvature. From (8.62), we see that R = 0 if and only if Ric = 0. Thus any two-dimensional spray with Ric = 0 and R =j:. 0 can not be induced by a Finsler metric. However, Finsler metrics of scalar curvature are quite special in higher dimensions. This leads to the notion of flag curvature. Let P C TxM be a tangent plane and y E P \ {0}. The pair { P, y} is called a flag in Tx M. Then P = span{y,u}, where u E Pis an arbitrary vector linearly independent ofy. Let L be a Finsler metric on an n-dimensional manifold M. It follows from (8.57) that the following quotient is independent of the choice of a particular u.

K(P, y)

:=

. gy(Ry(u), u) gy(y, y)gy(u, u)- gy(y, u)gy(y, u)

(8.64)

There The quantity K(P,y) is called the flag curvature of the flag {P,y}. is a special class of Finsler metrics L such that for any y E T M \ { 0}, the flag curvature K(P,y) is independent of the tangent planes P containing y. As matter of fact, this class is quite rich. For a Finsler surface (M, L), the tangent plane P at each point is the whole tangent space TxM. Thus by definition, F is contained in this class. In this case, K(y) := K(TxM, y) is a scalar function on T M \ {0}. We call it the Gauss curvature. Clearly, a Finsler metric is of scalar curvature K(y) if and only if for any y E TM \ {0} the flag curvature K(P,y) is independent of the tangent planes P containing y. In particular, Ry = 0 if and only if K(P, y) = 0. That is, a Finsler metric is of zero curvature if and only if it is R-flat. We have an analogue of Proposition 8.1.6 for Finsler metrics, which is due to L. Berwald

Proposition 8.2.4 ([Bw2]) A Finsler metric is locally Minkowskian if and only if B = 0 and R = 0.

Proof Let (M, L) be a Finsler space. Assume that L is locally Minkowskian. Then there is a local coordinate system (xi, yi) in which Gi = 0, hence B = 0 and R = 0. Conversely, if B = 0 and R = 0, then qk = qk(x) are functions of x only. The Berwald connection \7 is affine with vanishing curvature. Therefore, at every point, there is a local coordinate system in which qk = 0. Hence Gi = 0. By (4.29), we have (8.65) Differentiating (8.65) with respect to ym gives (8.66) Contracting (8.66) with y 1 yields

2Lxrn = 0.

8.2. RIEMANN CURVATURE OF FINSLER METRICS Thus

121

IJ

L(yl' ... 'yn) := L(yi a~i

is independent of x, i.e., Lis locally Minkowskian. We have proved the proposition. Q.E.D.

Remark 8.2.5 E. Cartan mentioned Proposition 8.2.4 in his monograph (Ca]. One can find a proof in (Nu] . See also (BaChSh1] .

Example 8.2.2 Consider the following Finsler metric

L = [ 3a(x, z)uw 2 + b(y)v 3]

2/3

.

According to Example 6.1.3, L is a Berwald metric. The coefficients of the Riemann curvature Rare given by

Rl1 - -cp uw, Ri 3

Rl where

1

= 0,

= 2'P w ·-

2

R~ R~

'

'P .- axz

Thus R

R~

(

X, Z

)

= 0, = 0,

= 0,

Rl3-- 'P u2 ' R~ 3

R3

=0

= - 21 cp uw,

ax(x, z) az(x, z) ( ) ( ) . ax,z ax,z

-

= 0 if and only if cp = 0.

Definition 8.2.6 Let L be a Finsler metric on n-manifold M. L is called an Einstein metric if there is a constant .\ such that the Ricci scalar R := n~l Ric satisfies (8.67) yETM. R(y) = .\L(y),

L is said to be Ricci-constant if R ii .·-- R x · 1

-

NiR 0· i y 1· -

(8.68)

L is said to be weakly Ricci-constant if (8.69)

It follows from Lemma 4.2.2 that every Einstein metric is Ricci-constant.

CHAPTER 8. RIEMANN CURVATURE

122

Example 8.2.3 Consider the following two-dimensional spray G

a = ua- + va- - 3v)u 2 + v 2ax

ay

au

-u 4 + u 2v 2 + 2v 4 a . u.,Ju2 + v2 av

(8.70)

By (8.14), a direct computation yields

~ ( 3u 2 + 2v 2 ) ( ~) 2 2 3(3u2 + 2v2 )(v) -2 ~ .

Thus the Ricci curvature

R

= R~ + R~ = 0.

By the above argument, we conclude that this spray can not be induced by any singular Finsler metric.

Example 8.2.4 Consider the following affine spray on R2

a

G := u ax

a 2a 2a + v ay - cjJ(x, y)u au - '1/J(x, y)v av .

(

8.71

)

Using (8.14), one obtains

R} R~

R~ R~

-uv¢y(x , y), -uv'I/Jx(x, y), v 2 '1/Jx(x, y), u 2 ¢;y(x, y).

The Ricci curvature is given by

(8. 72) Assume that ¢; and '1/J are related by

cPy(x , y) = -'1/Jx(x,y) For example,

'1/J

# 0. 1

(8. 73)

2

--x - xy . 2 Let f(z) be a non-trivial analytic function on 4_

(8.78)

= G1 and H = G2 of the spray induced by L are given G H

CHAPTER 8. RIEMANN CURVATURE

124

Thus L is a Berwald metric. By (8.42) or (8.44) , we immediately obtain a very simple formula for the Ricci scalar R

Thus R

=

= 0 if and only if Px y = 0.

Pxy

( ) uv. a1+a

In this case, L is locally Minkowskian.

= L(x,y,u , v)

Example 8.2.7 Let L U C R 2 in the form

L

(8. 79) U

be a Finsler metric on an open subset

= e 2 p(x,y)v 2 exp ( 2a;),

(8 .80)

where a is a constant. The main scalar of L satisfies

(8.81) The spray coefficients G = G 1 and H = G2 of the spray induced by L are given by

G H Thus Lis a Berwald metric. By (8.42) or (8.44), we immediately obtain a very simple formula for the Ricci scalar

(8.82) Thus R = 0 if and only if Pxx = 0. In this case, L is locally Minkowskian.

U

There are many two-dimensional Finsler metrics with K = 0 which are not locally Minkowskian.

Proposition 8.2.7 Let L following form

= L(x,y,u,v) L

be a Finsler metric on U C R 2 in the

= [u¢ ( x, y, ~)] 2 ,

where ¢ = ¢(x, y , 0 is a positive function with ¢€€ ::J 0. Suppose that¢ satisfies

are equal in the direction Y, Dy = Dy, i.e., their connection coefficients are related by Nj(Y)

= Nj(Y).

(8.96)

Let G and G be two Y-related sprays. Suppose that Y is geodesic with respect to G. It follows from (8.96) that (8.97) Thus Y is also geodesic with respect to

G.

The following theorem is very useful in comparison geometry.

Proposition 8.4.2 Let Y be a geodesic field on a spray space (M, G). Assume that a spray G on M is Y -related to G. Then for y = Yx E TxM (8.98) ,... ----------------------- -,

:/ / / : ,_' '

Ry(u) =;Ry(u)

u '

'

CHAPTER 8. RIEMANN CURVATUR E

130 Proof Observe that

8Gi (Y) oxk

(8.99)

and

This implies (8.100) Plugging (8.99) and (8.100) into (8.11) yields

Q.E.D. Now we prove the main result of this section. Propositio n 8.4.3 Let (M, L) be a Finsler space. For any y E TxM, (8.101) where R is the Riemann curvature of with Yx = y. Proof Let

g :=

gy and Y is a geodesic field of L

D denote the Levi-Civita connection of g.

By Lemma 7.1.4,

Dy = Dy. Thus Lis Y-related tog and (8.101) follows from Proposition 8.4.2.

(8.102) Q.E.D.

8.5. VARIATIONAL FORMULAS

131

Remark 8.4.4 Assuming that Proposition 8.2.1 holds for Riemannian metrics,

then it also holds for Finsler metrics by a simple argument using Proposition 8.4.3. Let (M, L) be a Finsler space. For a vector y E TxM \ {0} , extend y to a geodesic field Y in a neighborhood Ux. Let R denote the Riemann curvature of g := gy. By assumption, for any z E U and wE TzM \ {0}, the following holds

Restricting the above equation to x with w = y, we obtain (8.57).

8.5

Variational Formulas

For Finsler metrics, the geometric meaning of the Riemann curvature lies in the second variation formulas. In this section, we will give the second variational formulas for Finsler metrics. Let ( M, L) be a Finsler space. Consider the following variational problem £(c) := where c : [a, b] ---* M is a

lb

L(c(t))dt =Extremum,

coo curve.

Take a variation of c,

H: (-E,E) x [a,b]-* M with fixed endpoints, i.e. , H(O, t) = c(t),

H(u, a) = c(a),

Put

£(u) :=

lb L( 88~

H(u , b) = c(b) .

(u , t) )dt.

Assume c is a geodesic. Hence £'(0) = 0. Let V(t) .- ~~ (0, t). A direct computation yields the second variation formula (8.103) Thus, the sign of the Riemann curvature tells us some properties of [ nearby c. For example, if L is positive definite and the Riemann curvature satisfies

then for any variation H of c,

[" (0) 2: 0.

132

CHAPTER 8. RIEMANN CURVATURE

Assume that L is positive definite. Let F = variational problem of length .C(c) := where c : [a, b] --7 M is a speed curve of F with

coo

1b

VL.

We consider the following

F(c(t))dt =Extremum,

curve. Assume that c : [a, b]

F(c(t)) =A.

--7

M be a constant

=I 0.

Then the function

satisfies (8.104) where V_!_(t) is given by 1 Vj_(t) := V(t)- .x. 2 9c(t)(c(t), V(t)) c(t) . By (8.104), we can prove the Synge Theorem: for an even-dimensional oriented closed positive definite Finsler space (M, F), if the flag curvature is positive, then the fundamental group 1r1 (M) = 0. See [BaChSh1] .

Chapter 9

Structure Equations of Sprays Sprays are special vector fields on the tangent bundle. All of the quantities defined by a spray live on the tangent bundle. In previous chapters, we treat them as quantities on the base manifold by choosing a reference vector. To find the internal relationship among various quantities such as the Berwald curvature, Landsberg curvature and Riemann curvature, etc., one has to go up to the tangent bundle. It seems that exterior differential method is quite useful in computation for this purpose. We will employ it to do some complicated computations.

9.1

Berwald Connection Forms

Let M be ann-dimensional manifold. Let (xi, yi) be a standard local coordinate system in T M. The vertical tangent bundle of T M \ {0} is defined by VTM :=span{

8~ 1 ,

· · ·,

8~n}

and the canonical horizontal co-tangent bundle of T M is defined by 1-l*TM :=span{ dx 1 , · · · ,dxn }·

These are two special vector bundles over T M which are independent of any geometric structure. Now we consider a spray G = yi 0~, - 2Gi(y) 0~, on M. We are going to show that it determines the complements of VTM and 1-l*TM. Let 6 8 . 8 ·NJ Jxi .- 8xi - i 8yi'

133

134

CHAPTER 9. STRUCTURE EQUATIONS OF SPRAYS

· ac' where N] := ay' . Put HTM :=span{

6~ 1 ,

···,

!n },

6

One can easily show that 1iT M and V*T M are well-defined vector bundles over T M \ {0}. We obtain a direct decomposition for T (T M \ {0}) T(TM \ {0}) = HTM rJJ VTM, and a direct decomposition for T * (T M \ {0}) T*(TM \ {0}) = 1i*TM rJJ V*TM. There are two canonical sections of VT M and HT M. .f)

Y ·.- y' ~'

uy'

. 6

G ·.- y' ~· uX 2

(9.1)

In this sense, G is a horizontal vector field on T M \ { 0} .

TM

•y ....,.._______ _}-··· • ---) Sx ',__________...· G

The above setting is in a standard local coordinate system in TM. Now we turn to an arbitrary local (co-)frame to gain more freedom. Take an arbitrary local coframe {wi}f= 1 for 1i*T M and the corresponding coframe {wn+i}f= 1 for V*TM . Express them by (9.2) where (a}) is a local n x n matrix-valued function on TM \ {0}. Let (a}) - 1 . The dual frame by

{ei}~ 1

(b}) :=

for VTM and {en+di= 1 for HTM are given en+i

. 6

= b~~­ uxl

9.2. CURVATURE FORMS AND BIANCHI IDENTITIES

135

The Berwald connection can be defined as a linear connection on VT M. Let · - ak i {dbkj wj i .-

where qk

= 8~~~~k .

+ bmrk j ml d x 1} '

The Berwald connection \7 on VT M is defined by \7 X:= dXi 0 ei

+ Xiw/

0 ei,

(9.3)

where X= Xiei E C 00 (VTM). We call {w/}0= 1 the Berwald connection form with respect to {wi}j= 1 . The co-frame {wn+i}j= 1 can be expressed as

(9.4) where vi = ajyi are the coefficients of Y = viei.

9.2

Curvature Forms and Bianchi Identities

As we have seen, the Cartan connection and the Chern connection are defined only for Finsler spaces, while the Berwald connection is defined for general spray spaces. Therefore, we will use the Berwald connection as our main tool to study spray spaces and Finsler spaces. Let (M, G) be a spray space. Take an arbitrary local coframe {wi}j= 1 for 1/.*TM and the corresponding coframe {wn+i}j= 1 for V*TM. Let {wj} be the Berwald connection forms with respect to {wi}i=I· The symmetry qk = is equivalent to (9.5)

rL

Put (9.6) The matrix-valued 2-form fl = (fl/) is called the curvature form of the Berwald connection with respect to {ei} or {wi} . Differentiating (9.5) yields

Thus fl/ must be in the following form

nJ.i = ~2 R J_; kl wk 1\ wl where

Bi wk 1\ wn+l Jkl )

+ R/tk = 0, + Rki tj + R1 iik =

(9.8)

R/kt R/ kl

(9.7)

0

(9.9)

and (9.10)

CHAPTER 9. STRUCTURE EQUATIONS OF SPRAYS

136 Put

(9.11)

Differentiating (9.4) yields ni

= vlf! J i •

(9.12)

Thus we can express f!i in the following form (9.13) where

. viBijkl· i .= B kl

. viRj i kll R i kl .=

Put

i tR; tRi . R ik·=V kt=VV jkt ·

(9.14) (9.15)

As a matter of fact, the above quantities in (9.7) and (9.14) are not new. We will show that Ri i kt are exactly the coefficients of the Riemann curvature tensor given in (8.20) and B}kt are exactly the coefficients of the Berwald curvature given in (6.4). Moreover, BL1 = 0. First we prove the following

Lemma 9.2.1 d!!j

i

+ nj k 1\ wki

k

i

- wj 1\ nk = 0.

(9.16)

Proof The proof is direct, using (9.6).

dD/

- dw/ 1\ wki

+ w/ 1\ dwki

+ w/ 1\ w/) 1\ wki + w/ 1\ (Dkj + wk1 1\ w 1 i) -nj 1\ wki + wj k 1\ nki .

-(D/ k

Q.E.D. Define R/ kl;m and R/ kl ·m by

(9.17) Similarly, we define B}kt;m and B}kt ·m by

(9.18) From (9.16), one obtains the following Bianchi identities R/ kl;m

+ R/ lm;k + R /

mk;l = 0;

(9.19)

. i R j i kl ·m -- Bijml;k - B jkm;l'

(9.20)

B}kt·m = B}km·t·

(9.21)

9.3. R-QUADRATIC AND R-FLAT SPRAYS

137

Contracting (9.19) with vj, we obtain a Bianchi identity for R\1 = R/ klvj. (9.22) Contracting (9.22) with v 1 yields i

i

Rk;m - Rm;k

+ R i mk;lvl

(9.23)

= 0.

Let {wi := dxi, wn+i := Jyi} be the natural local coframe for 1l*T M EB V*TM. We obtain f!_;

J

ari ari ari11 - +Nm~ -rmri }dxk l\dx 1 - ~dxk l\by 1 . = {oyl Jk ml l oym oxk

Thus

.

R/kl

=

1, L is a Riemannian metric. In either case, the Gauss curvature K at p = (x, y) is given by K=

f"(x)

2"

f(x) [1- f'(x)

2]

(i) Take

f(x)=~. The resulting metric has Gauss curvature K (ii) Take

f(x) =ax+ b,

= 1. a f ±1.

(11.5)

155

11.1. FINSLER METRIC OF SCALAR CURVATURES

The resulting metric has Gauss curvature K = 0. Note that when a= ±1, the induced metric on M 2 is degenerate.

(iii) Take a function f satisfying (11.6)

with

f(O) = 1,

j'(O)

=I ±l.

The resulting metric has Gauss curvature K = -1. The above surfaces in R 3 look like

K= 1

K=O

K= -1

We see that a surface of constant curvature K = 1 in (R3 , g) is curved as a surface of curvature K = -1 in the Euclidean space lR 3 . Compare Example 11.3.3 below. Let

f(x)

= VxZ=l,

(11. 7)

One can easily verify that f(x) is a solution of (11.6). The induced metric Lon the surface is given by

L also has Gauss curvature K

= -1.

This surface in R 3 looks like

CHAPTER 11 . FINSLER SPACES OF SCALAR CURVATURE

156

This hyperbolic surface was discovered by Beltrami in 1868. Example 11.1.2 Consider a Randers metric F = o: + (3 on a manifold M, where o: is a Riemannian metric and (3 is a close 1-form. According to Example 8.2.1, F is of scalar curvature if and only if o: is of scalar curvature (constant U curvature when n =dim M 2: 3) . Assume that a Finsler metric L is of scalar curvature >. = >.(y) . So the Riemann curvature is in the form (11.1). In local coordinates, (11.1) is expressed as follows (11.8) For simplicity, let

and Let and

t s 1 h ij := 9ii - ILL 4 L ·i ·i = 9ii - £9isY 9itY ,

(11.9)

s i 1 J:i >i - 1L L ·jY i -_ uj ·- giph jp -_ uj hij .- £9isY Y · 2

We are going to apply Bianchi identities (9.19)-(9.23) to study a Finsler metrics of scalar curvature >. = >.(y). We show that in dimension 2: 3, if, in addition, >. = >.(x) is a scalar function on M, then>. must be a constant. Plugging (11.8) into (8.17) gives R j i kl

>i} >i - L ·kU! >..i {L ·lUk >.·i·k Lhi[ + 2 Aft Lhik - -3-3.Ji +~{L ·J k 3

>..k - gJ·kyi- ~L J 2 1Ji} 3 {L ·J·Jf- gJ·tYi- ~L J 2 .kJi}-

+>-{gj!Jk- 9jkJf}

°

(11.10)

11.1. FINSLER METRIC OF SCALAR CURVATURES

157

Contracting (11.10) with yi yields (11.11) Differentiating (11.11) along the direction ym o:~ yields (11.12) Differentiating (11.8) along the direction 0 ~ 1 yields (11.13) Plugging (11.12) and (11.13) into (9.23), we obtain \tL h1- \kL ht-

~\mYm{ L.t81- L.k8;} >...t;mYm L hi _ Ak;mYm L hi.

3

k

3

l

(11.14)

Now we assume that >.. = >..(x) is a function of x only. Then >...i = 0 and (11.14) simplifies to (11.15)

Suppose that n =dim M 2: 3. Fix a vector y E TxM with L(y) =/= 0. There are two non-parallel vectors u, v E TxM which are gy-orthogonal to y, i.e., (11.16) Contracting (11.15) with uk and v1 yields i (v l \tL) u i = (u k >..;kL) v.

We conclude that for u E TxM with gy(y, u) = 0, the following holds (11.17) By the continuity of>.., we conclude that (11.17) holds for any u E TxM. This proves the following Proposition 11.1.1 Let (M,L) be a Finsler space of dimension n 2:3. If Lis of scalar curvature>..= >..(x) depending on position only, then >..(x) =constant.

158

CHAPTER 11. FINSLER SPACES OF SCALAR CURVATURE

Next, we are going to give a brief argument on Numata's theorem that any Landsberg metric of scalar curvature in dimension 2 3 must be Riemannian on the open subset of M where the curvature scalar .\(y) f:. 0. Contracting (11.10) with y 1 yields .

R/ktY

l

= (11 .18)

Plugging (11.18) into (10.13), one obtains 1 = - .\.i L h ·k L t]·k·tY • 3 J

-

.\-i 3 L h •·k -

3 L h t]· · - .\LC.t]·k.

.\.k

(11.19)

From now on, we assume that L is a Landsberg metric. (11.19) reduces to Cijk =- 3\ { .\.ihJk

+ .\.Jhik + .\.khiJ } ·

(11.20)

Contracting (11.20) with gik yields

ci

n+l

= -~.A.i,

(11.21)

where Ci := gikCiik· Substituting (11.21) back to (11.20), we obtain (11.22)

Thus L is C-reducible in the sense of Matsumoto (Mal]. In other words, any Landsberg metric of scalar curvature .\(y) f:. 0 must be C-reducible. When n 2 3, L is actually Riemannian. Proposition 11.1.2 ((Nu]) Let (M, L) be a Landsberg space of dimension 2 3. Suppose that L is of scalar curvature .\ f:. 0. Then L is Riemannian.

S. Numata first proved the proposition for Berwald spaces of scalar curvature. Then he proved the proposition for Landsberg space of scalar curvature based on a result of M. Matsumoto on C-reducible spaces (Mal]. What happens when R = 0 for a Finsler metric L in Proposition 11.1.2 ? According to Theorem 10.3.7, if, in addition, L is positively complete and positive definite, then it is locally Minkowskian, provided that C is bounded. Finsler metrics constructed based on Proposition 8.2.7 or Theorem 10.3.5 are in general R-flat non-Landsbergian metrics. A natural question is whether or not there is a Landsberg metric on Rn with R = 0, which is not locally Minkowskian.

11.1 . FINSLER METRIC OF SCALAR CURVATURES

159

Note that all two-dimensional Finsler metrics are of scalar curvature. In dimension two, Numata's theorem might not be true. Assume that L a twodimensional Berwald metric. Then the Gauss curvature K satisfies a special equation along the indicatrix at each point. If L is positive definite with K i= 0, one can see that the main scalar must be zero. Thus, L is Riemannian. This fact is due to Z. Szabo [Sz1]. Refer to Proposition 6.1.6. See also [BaChSh1] for a proof. Note that if K = 0, then L is locally Minkowskian. Anyway, we still do not know whether or not there is a (two-dimensional) Landsberg metric which is not Berwaldian ! Is there a non-trivial two-dimensional R-fiat Landsberg metric? We do not have any clue yet. Let L be a Finsler metric of scalar curvature >.(y). Differentiating (11.10) with respect to ym gives a formula for kl·m expressed in terms of>. and its kl·m = B}mt;k - B}km;l with yk, one obtains derivatives. Contracting

R/

R/

2>.CjtmY;

(11.23)

>.-i { 1 L !d ,; 2 ;} -3 2 ·lUm + 21 L ·mUt9tmY >..t { 1 L Ji -3 2 ·j m + 21 L ·m 6j; - >..m 3

2gjmY

i}

{~L ·Jf + ~L tbi- 2gJ tYi} 2 ·J 2 . J

- >.·i·m L hti - >.·i·l L hi - >..t·m L hi 3 3 m 3 J'

(11.24)

where hj = Jj- AL.jyi. It follows from (11.24) that (11.25) One can also derive (11.24) and (11.25) from (9.50) and (9.51) respectively by taking R =>.Land Tk = -~>.L.k. Assume that L is of scalar curvature >. = >.(x) depending on x only (Note: >.=constant when dim> 2 by Proposition 11.1.1). In this case, (11.19), (11.24) and (11.25) simplify to Lijk;mYm

->.LC;jk,

(11.26)

Eij;mYm Bi m jkl;mY

0,

(11.27)

2>.CjktYi·

(11.28)

The above identities will be used in the following section.

160

11.2

CHAPTER 11. FINSLER SPACES OF SCALAR CURVATURE

Finsler Metrics of Constant Curvature

The local structures of Finsler metrics of constant curvature have not been completely understood. The Funk metric F on a strongly convex domain n C Rn is non-reversible with constant curvature K = -1/4. The Funk metric F is positively complete, while its reserve F(y) := F( -y) is negatively complete. The Klein metric F := ~(P +F) is reversible and complete with constant curvature K = 1. The Bryant metrics on sn are non-reversible with constant curvature K = 1. All of the above mentioned Finsler metrics are locally projectively fiat. Namely, at every point, there is a local coordinate system in which geodesics are straight lines. See Chapter 13 for more discussion on the projective geometry of Finsler spaces. Let us take a look at a Finsler metric L = L(x, y, u, v) on an open subset U C R 2 in the form where¢>= ¢>(x,y,~) is a function on U x R satisfying 1/>t,f, =P 0. According to Proposition 8.2.7, if¢> satisfies (8.83), i.e., (11.29) then the Gauss curvature K

= 0.

For a solution ¢> of (11.29), if

is not a polynomial of degree three or less in ~' then L is not projectively fiat. There are many solutions ¢satisfying the above conditions. Thus there are many non-projectively fiat R-fiat Finsler metrics. But it is not clear whether or not the resulting Finsler metrics are positive definite and regular in all directions. See Remark 13.6.7 below. Recently, Bao and Shen have constructed a family of Randers metrics on S3 of constant curvature K = 1. These metrics are not locally projectively fiat. The details are given in [BaSh2] and Example 11.2.1 below. The classification of Finsler metrics of constant curvature is far from being done. In this section, we are going to study the Cartan torsion, the Landsberg curvature and the Berwald curvature along geodesics in a Finsler space of constant curvature .A. We will also discuss some interesting examples. For a number .A E R, let

S>, (t)

:

{

sin(~t) ~

~inh(vCXt) v=x

if .A> 0 if .A= 0 if .A< 0.

(11.30)

11.2. FINSLER METRICS OF CONSTANT CURVATURE

c >. (t) :

cos( v'Xt) { 1 cosh( vC>:t)

=

if A > 0 if A = 0 if A < 0.

161

(11.31)

Note that s~(t) = C>.(t),

Thus both S>.(t) and C>.(t) satisfy the following equation

y"(t)

+ Ay(t)

= 0.

Let (M , L) be a Finsler space. Take a geodesic c(t). By Lemma 4.2.2,

:t

[L(c)J = G(L) = 0.

o

Thus, L(c(t)) = is a constant. But this constant might be zero or negative in the case when Lis not positive definite. Take an arbitrary vector field V = V(t) along c. Let C(t): L(t): E(t):

=

B(t) :

=

Ct(t)(V(t), V(t), V(t)), Lt(t)(V(t), V(t) , V(t)), E c(t )(V(t) , V(t)), Bc(t)(V(t) , V(t), V(t)).

C(t) and L(t) are functions oft and B(t) is a vector field along c. Our first result is as follows. Proposition 11.2.1 Assume that (M, L) is a Finster space with constant curvature A. Then for any geodesic c with L(c) = o and any vector field V(t) along c, C(t) L(t) E(t)

as>.Jt) + bc>.Jt), ac>.Jt)- A0 bs>.Jt), e,

B(t)

(11.32) (11.33) (11.34) (11.35)

Here a, b, e are constants, A0 := AO and E(t) is a parallel vector field along c perpendicular to c( t) with respect to 9c(t) . Proof It follows from (11.26) that (11.36) For simplicity, let

B'(t)

:=

D.;B(t).

162

CHAPTER 11. FINSLER SPACES OF SCALAR CURVATURE

By (10.16) , (11.28), (11.27) and (11.36), we obtain the following system C'(t) L'(t) E'(t) B'(t)

L(t) , - . \ 0 C(t), 0, 2.\C(t)c(t).

(11.40)

The proposition follows from the above equations.

Q.E.D .

(11.37) (11.38) (11.39)

From Proposition 11.2.1, we immediately obtain the following Theorem 11.2.2 ([AZ2]) Let (M , F) be a complete positive definite Finsler space of constant curvature .\ < 0. If C does not grow exponentially, then F is Riemannian. In particular, any compact positive definite Finsler space (M, F) with negative constant curvature must be Riemannian.

Proof Take an arbitrary geodesic c defined on ( -oo, oo) with b = F(c) > 0 and an arbitrary parallel vector field V(t) along c. By assumption A0 = .\b < 0. Define C(t) as above. By Proposition 11.2.1, we obtain C(t)

= as>.Jt) + bc>.Jt).

Clearly C(t) is unbounded on ( -oo, oo) if a IICII This implies that

=

f= 0 or b f= 0. By assumption,

sup ICv(v, v , vjl < oo. yf.O,vETM [gy(v,v)]2

IC(t)l :::; IICIIQ~ < oo,

where Q = gc(V(t), V(t)) is a positive constant by Lemma 7.3.2, since V(t) f= 0 is parallel along c. We conclude that a= b = 0 and C(t) = 0. Since the geodesic c(t) and the parallel vector field V(t) along care taken arbitrarily, we conclude that C = 0, i.e., L is Riemannian. Q.E.D . From (11.34) , we see that on a Finsler space of constant curvature, the mean Berwald curvature is constant along geodesics. A natural question arises: is there any non-trivial Finsler metrics of constant curvature with vanishing mean Berwald curvature ? So far, we have only found one example. First, by Example 5.2.3, there exist a lots of Randers metrics on S3 with S = 0. By (6.13), such Randers metrics must satisfy E = 0. From this family of Randers metrics with S = 0, we have found a special family of Randers metrics with constant curvature K = 1. See Example 11.2.1 below . Example 11.2.1 ([BaSh2]) On the Lie group Sp(1) = S3 , there are three lin2 ) ( 3 satisfying early independent right invariant 1-forms

e)(

(11.41)

11.2. FINSLER METRICS OF CONSTANT CURVATURE

163

Let (21

and (/

= -(/

:=

(3)

(31

We obtain d(i

:=

-(2)

(3 2

:=

(1

= (i A(/.

(11.42)

Fix an arbitrary constant k 2 1. Let {8 1 := Vk( 1 ,B 2 := ( 2 ,8 3 := ( 3 } and {b 1 ,b 2 ,b 3 } denote the frame which is dual to {8 1 ,8 2 ,8 3 }. Put (11.43) and

B/ = -B/. It follows from (11.42) that (11.44)

dBi = Bi A B/

Define a Riemannian metric o: and a 1-form /] on S3 by

/](y) := where y = ub 1 + vb 2 b2 B2 + b3 B3 , where

+ wb 3 .

Jk=l .Jk u,

Warning: o: depends on k ! Write /] :=

b1B 1

+

The covariant derivatives of/] are defined by db t.- b·B J i i -· - . b·t)·Bi .

An easy computation gives

bu = 0, b21 b31

= 0, = 0,

= 0,

b12 b22

b32

= 0, =

b13

b23 =

-Jk-=1,

= 0,

-Jk=l b33

= 0.

Now we treat o: as a special Finsler metric, hence we go to the slit tangent bundle TS 3 \ {0}. Let · -- 7r *B j i . wj i .

The same relationship between from (11.44) that

wi

and

w/ as in (11.43) still holds.

It follows (11.45)

Note that {w/} are just the Berwald connection forms of o: on TS 3 with respect to {w1 , w2 , w3 }. In order to define the covariant derivatives of a tensor on

CHAPTER 11 . FINSLER SPACES OF SCALAR CURVATURE

164

TS 3 \ {0} with respect to the Berwald connection of o:, we need to expand {w 1 w 2 w3 } to a coframe {w 1 w2 w3 w 4 w5 w6 } as follows

'

'

'

'

'

'

'

i = 1,2,3,

where (y 1,y 2,y 3) = (u,v,w) are functions on TS 3 determined by y = ub 1 + wb 3 . An easy computation gives

+

vbz

w4

:

w5

:

Vk w w2 + Vk v w3 dv + ( ~ - Fk) w w 1 - Vk u w3 du -

=

w6 :

dw - (

~ - Fk)

v w1

+ Vk u w 2.

Since o: is Riemannian, we have

f!/ := dw/- wj .

.

k

.

1\ wk" =

1

2R/ klw

k

l

1\ w.

The Riemann curvature coefficients R% := R/ klyiy 1 must be functions of (y 1 , y 2, y 3 ) = (u, v, w) only. R~ R~ = -kuv

Rr = -kuw

R~ = k(u 2

= -kuv ,

+ w 2 ) - 4(k- 1)w2 ,

R~ = -(4- 3k)vw

Let F := o: F is a Randers metric on satisfies

83 .

R~

R~ =

k(u 2

= -kuw R~ = -(4- 3k)vw

+ v 2 ) - 4(k- 1)v2 .

+ f3

By Example 5.2.3 and (6.13), we see that F

s=

0,

E =0.

Thus F is weakly Berwaldian. But it is not Berwaldian, nor Landsbergian. In what follows, we are going to compute the Riemann curvature of F. Let {ei}~=l be the frame dual to {wi}~=l· Note that {e 1,e 2 ,e3} is the horizontal lift of {b 1, b 2 , b 3 } and {e 4, e 5, e6} is the set of vertical tangent vectors on TS 3.

The spray G of o: is a horizontal vector field in the form

It follows from (5.28) that the spray

G ofF is in the form

G =G-2Q,

165

11 .2. FINSLER METRICS OF CONSTANT CURVATURE where (y 1 , y 2 , y 3 ) = (u, v, w). A direct computation gives

(11.46)

Note that Qi are functions of (u, v, w) only. By Lemma 8.1.4, the Riemann curvature coefficients of G are related to that ofG by (11.47) where

Hk

:= 2Q:k - Q\;jYj

+ 2QjQij·k- QijQ~k·

(11.48)

Here the covariant derivatives of Q with respect to a are defined by

dQi ·k

+ Qj·k W·iJ

_ Qi.w j = Qi wl ·k;l •J k

+ Qi·k·l wn+l.

By a direct computation, we obtain

Q\ =0, Q~1 = 0, Q~1 = 0,

-Jk=l wau, Q 31 = Jk=1 vau,

Q~1 =

Q\ =0, Q 22 = -Jk=l wav, Q32 =

Q 23

= -Jk=l (a+ waw),

Jk=1 (a + vav),

Q 33 =

Vk-=-1 vaw.

Similarly, we obtain all formulas for Qik·l and Qik·L · We omit the details here. ' ' Plugging them into (11.48) yields

Jkv'k-=1(v

Hf

+w 2

)ua-

1

-JkJk=I u 2 va- 1 -Jkv'k=l u 2 wa- 1 -(k- 1)uv- Jkv'k=l( 2u 2 + v2 + w 2 )va- 1

H~

Hi H? Hi

2

=

(k- 1) ( u 2

+ 4w 2 ) + 2Vkv'k=l ua

JkJk=I uvwa- 1

H~

-4(k- 1)vw-

Hf

-(k- 1)uw- Vkv'k=1(2u 2

H~

-4(k- 1)vw-

HJ

(k -1) ( u2 + 4v 2 ) + 2Vkv'k=l ua

+ v2 + w 2 )wa- 1

JkJk=I uvwa- 1

CHAPTER 11 . FINSLER SPACES OF SCALAR CURVATURE

166 We add

Ri and Hk together and find a simple formula for Ri,

where (y 1 , y 2 , y 3 ) = (u, v, w). Thus F is of constant curvature K = k 2: 1. Normalizing F, we obtain a family of Randers metrics F' = ~F on S3 with U constant curvature K = 1. There are some other interesting Randers metrics of constant curvature. Example 11.2.2 On the unit ball Bn in IRn, the Funk metric is given by

IYI 2 - (1xi 2 1YI 2 - (x, Y) 2 ) F(y) := ~---,--1--...,..lx--:-:12~--

+

(x, y) 1 -lxl 2 ·

where I · I and (,) denote the standard Euclidean norm and inner product in Rn. The Funk metric F has constant curvature K = - 1/4. Moreover, F is only positively complete. The Cartan torsion C is bounded too. More precisely, at any point x E Bn , IICII x :=

sup

y ,vET. M

ICy(v, v, v)l

3

[gy(v,v)]~ ~ v'2.

Does this contradict Theorem 11.2.2? It is then a natural problem to classify all Randers metrics with constant

curvature. This problem was first studied by M. Matsumoto [Ma2], then by Shibata-Shimada-A zuma-Yasuda [SSAY]. Finally, Yasuda-Shimada [YaSh] in 1977 successfully found a set of sufficient and necessary conditions on (3 for F with constant curvature. Surely, the computation is very complicated. Many years later, Shibata-Kitayama [ShKi] and Matsumoto [Ma6] verified their result in a different approach for Randers metrics of constant curvature. Inspired by Yasuda-Shimada's result, Bao-Shen find the above family of Randers metrics on S3 with K = 1. We continue to study Finsler spaces of constant curvature. Let (M, L) be a Finsler space of constant curvature >.. (11.18) simplifies to (11.49) Plugging (11.36) and (11.49) into (10.35), (10.36) , (10.40) and (10.41) yield

(11.50)

11 .3. RIEMANNIAN SPACES OF CONSTANT CURVATURE

+ Lijk·l, >-{ Ljklgis + Liklgjs + Li1Wks }y• ->.LCijk ;l + >.LLijkl ,

Cijk·l;mYm

Lijk ·l ;m Ym

(11 .51)

-Cijk;l

Lijk;l ;m Ym

=

->.LCijk·l - Lijk ;l - 2>.CijkglsY

8

167

(11.52) •

(11.53)

Let c be a geodesic with

o := L(c) and V(t) , W(t) parallel vector fields along c. Define C(t), C(t) , L(t) and L(t) as in (10.49)-(10.52). When V(t) = c(t) or W(t) = c(t), the above functions take special forms . For simplicity, we assume that gc(t) (

c(t), V(t)) =

o=

gc(t) (

c(t), W(t)).

It follows from (11.50)-(11.53) that

L' (t)

+ L(t), -C(t) + L(t), ->. C(t) + >-aL(t),

:L' (t)

->. 0 C(t)-

C'(t) C'(t)

>-aC(t) 0

L(t),

(11.54) (11.55) (11.56) (11.57)

where A0 :=>.b. Luckily, the above system is solvable. The interesting case is when A0 Assume that A0 = 1. From (11.54)-(11.57), we obtain C(t) C(t) L(t) L(t)

a sin(2t) + (3 cos(2t) + c, -(3 sin(2t) +a cos(2t) + d, -(3 sin(2t) +a cos(2t) - d, -a sin(2t) - (3 cos(2t) + c,

= 1.

(11.58) (11.59) (11.60) (11.61)

where a, (3, c and dare constants. From the above identities, we see that C(t), et al are periodic functions of period 71' (not 271') along geodesics for any gc(t)-orthogonal parallel vector field V(t).

11.3

Riemannian Spaces of Constant Curvature

One can easily show that for a Riemannian metric g on a manifold M, the flag curvature K(P, y) of a flag {P, y} C TxM is independent of the direction yEP. Thus K(P, y) = K(P) depends only on the tangent plane PC TxM. In this case, we call K(P) the sectional curvature of the section P. By a similar argument, one can also show that if g is of scalar curvature >. = >.(y), then

168

CHAPTER 11 . FINSLER SPACES OF SCALAR CURVATURE

A = A(x) is a function of x E M only. In this case, the sectional curvature is constant at each point, namely, K(P) = A(x), Further, in dimension n ~ 3, the scalar function A = constant. Riemannian metrics of constant curvature have been completely classified. In this section, we will discuss some examples of Riemannian metrics of constant curvature. First let us take a look at the following Example 11.3.1 (Klein Disk) Consider the Klein metric on the unit ball :IBn in :!Rn (see Section 2.3)

where (,) and I · I denote the standard inner product and Euclidean norm in :!Rn. By a direct computation, we obtain

2 8gjl _ 8gjk 8xl 8xk

=

38jtXk

+ 8ktXj

4xjxkxt

(1 -lxl2)2 + (1 - lxl2)3.

(11.62)

Plugging (11.62) into (8.85) gives (11.63)

We obtain the spray of L.

G - '~- 2 (x,y) '~ 1 - lxl2 y 8y• . - y ax• Clearly, the geodesics of L are straight lines in :IBn.

11.3. RIEMANNIAN SPACES OF CONSTANT CURVATURE

169

Plugging P = 1(~i!'f2 into (9.43) , we can obtain

1 "] . . 2,LykY' Ric. = - [LJJ.,-

(11.64)

Namely, L has constant curvature K(P) = - 1.

Example 11.3.2 Consider the following Riemannian metric on Rn

A direct computation yields

G- ,.!..._ - y &xi

+

,.!..._ 2 (x, y) 1 + lxl2 y &x• ·

(11.65)

Plugging P = - 1~1![2 into (9.43), we obtain .

.

1

RJ., = LJJ.,- 2,Lyky'.

.

(11.66)

Namely, L has constant curvature K(P) = 1. One of the most important theorems in Riemannian geometry is the classification of simply connected Riemannian spaces with constant sectional curvature. This is due to E. Cartan. Theorem 11.3.1 For any>. E {-1,0, 1} and n 2: 2, there is an unique complete simply connected n -dimensional Riemannian space with constant sectional curvature >..

170

CHAPTER 11. FINSLER SPACES OF SCALAR CURVATURE

Complete simply connected Riemannian spaces with constant sectional curvature are called the (Riemannian) space forms. We will describe them below in details. The space form with A = 1 is the standard unit sphere §n = (Sn, gsn) in !Rn+I_ Let {p, q} denote the north pole and the south pole of sn respectively. gn \ {p, q} >:::! (0, 1r) X gn- 1 . gsn on gn \ {p, q} is expressed by gsn = dt ® dt

+ sin 2 t gsn-1.

(11.67)

The space form with A = 0 is just the Euclidean space !Rn = (Rn, gR") Topologically Rn \ {0} >:::! (0, 00) X sn- 1 . gRn on Rn \ {0} is expressed by (11.68) gHn

The space form with A = -1 is called the hyperbolic space lHin on Rn \ {0} >:::! (0, oo) X sn- 1 is expressed by gHn = dt 18) dt

+ sinh2(t)gsn-t .

= (Rn, gH"). (11.69)

Example 11.3.3 Consider the following Riemannian metric on a surface L := a(x , y) u 2

+ c(x,y)

(11 .70)

v2 .

An easy computation yields

vac - 2vac [(~)

1 K- _ _

x

vac

+ (~) ] y

·

Take a look at a special surface t.p: (a , b) x S1 -+ R 3 defined by

t.p(x,y)

:=

(!(x)cosy, f(x)siny, x).

(11.71)

11 .3. RIEMANNIAN SPACES OF CONSTANT CURVATURE

171

The induced Riemannian metric is given by L =

(1 + f'(x) )u 2

2

+ f(x) 2 v 2

It follows from (11.71) that

K=-

f"(x) 2' f(x) [ 1 + f'(x)2 J

(a) Take

f(x)=~. Then K = 1. This is the standard unit sphere § 2 in :IR 3 . (b) Take

f(x) =ax+ b.

Then K = 0. The surface is a cone in :IR 3 . (c) Take f(x) such that

Then K = -1.

K= 1

K

K=O

= -1

There are many incomplete Riemannian metrics of constant curvature. Example 11.3.4 Take family of Riemannian metrics 9c := dt l8l dt

+ c2 sin 2 (t)g

0 ,

on (0, 1r) x S1 , where g0 = ds l8l ds denotes the canonical metric on S1 . One can U verify that 9c has constant curvature >. = 1 for any c > 0.

Chapter 12

Projective Geometry Two sprays G and G on a manifold are said to be pointwise projectively related if they have the same geodesics as point sets. For any geodesic c(t) of G, there is an orientation-preserving reparameterization t = t(s) such that c(s) := c(t(s)) is a geodesic of G, and vice versa. In this chapter, we will show that two sprays G and G on a manifold are pointwise projectively related if and only if there is a scalar function P on T M \ { 0} such that

G= G-2PY .

(12.1)

Then we prove the Rapcsak theorem on projectively related Finsler metrics. This remarkable theorem plays an important role in the projective geometry of Finsler spaces. See [Th3] for a systematic survey on the early development in this field.

12.1

Projectively Related Sprays

Let G(y) = yi 8~. - 2Gi(y) 88 , be a spray on a manifold M. Geodesics are locally characterized by the following system 2 i

~ 2G;(dc) = 0 dt 2 + dt , where (d(t)) denote the coordinates of c(t). Consider another spray G 2Gi(y) 8~. on M, where

(12.2)

= yi 8~.(12.3)

The scalar function P must satisfy the following homogeneity condition

P(>..y) = >..P(y), 173

>.. > 0.

(12.4)

174

CHAPTER 12. PROJECTIVE GEOMETRY

Assume that c(t) is a geodesic of G, hence the coordinates (ci(t)) of c(t) satisfy (12.2) . Take a new parameter s determined by ds dt

d 2 s = 2P(dc) ds dt 2 dt dt'

Then e(s)

:=

> O.

e(t(s)) satisfies

-2Gi(!k) dt

ds _ dt

dci d 2 s dt dt2

Thus e(s) is a geodesic of G. This implies that G and G are pointwise projectively related. The converse is true too. Assume that sprays G and G have the same geodesics as point sets. Take an arbitrary geodesic e in M. Let t and s denote the geodesic parameters of e with respect toG and G respectively. Parameterize c by a common parameter u. By (12.2) we obtain

d2ci du 2

d2 ei du 2

where

+ +

2Gi (de) = A.( ) dei du

2Gi ( de ) du

(12.5)

'~' u du '

= A..(

) dei

(12.6)

'~' u du '

d2 u

¢(u) :=

(~2'

The difference of (12 .5) and (12.6) is

2C;i ( de ) _ 20 i ( de ) du

du

= (J> _ 4>) dei . du

Since the above equations hold for any geodesic, we conclude that (12.3) holds for some function P on T M satisfying (12.4). We have proved the following

Lemma 12.1.1 Let (M, G) be a spray space. A spray G is pointwise projective to G if and only if there is a scalar function P on T M \ {0} such that

G = G-2PY.

175

12.1. PROJECTIVE LY RELATED SPRAYS

Consider a spray G on a manifold M. Fix a standard local coordinate system (xi, yi) in T M and write G = yi a~' - 2Gi(y) a~'. Let

a= 2,···,n,

(12. 7)

(;1 := 0, It is easy to verify that the local spray G := yi a~' - 2Gi(y) a~' is pointwise projective to G. Construct another local spray as follows

.a

.

a

(12.8)

II:= y"~- 2II'(y)~, uy" ux" where

.

.

.

1

II'(y) := G'(y)- -Nm(y) y".

n+

1

m

(12.9)

Clearly, II is also pointwise projective to G. II is called the local projective spray associated with G. II has the following important property: If a spray G is pointwise projective to G , then the local projective spray fi associated with G is equal to II in the same standard local coordinate system ! More important, any (local) invariant expressed in terms of IIi is a (local) projective invariant. However , "projective invariants" defined by II might not be globally defined. To avoid this problem, we need an additional structure on the manifold, that is, a volume form. Fix a volume form df-L = O'(x)dx 1 · · · dxn on M. For a spray G on (M, df-L), let S denote the S-curvature of ( G, df-L). Define

-

2S n+1

G:=G+-- Y.

(12.10)

G is a globally defined spray on M ! We call G the projective spray associated with G on (M , dJ.L). In a standard local coordinate system (xi,yi) in TM, G = yi a~' - 2Gi(y) a~' is given by (;i(y) := G;(y) _ S(y) yi.

n+1

(12.11)

By the definition of the S-curvature, we have

(;i( ) =IIi( ) + _1_ym aO' i n + 1 (]' axm y ) y y

(12.12)

where IIi are given in (12 .9). Thus G depends only on the pointwise projective class of G for a fixed volume form df-L. This implies that any invariant defined

176

CHAPTER 12. PROJECTIVE GEOMETRY

by G is a globally defined projective invariant. However, the invariants defined by G possibly depend on the volume form df.L too. As long as the invariant is independent of the volume form df.L, it is a globally defined projective invariant of G . Example 12.1.1 Let ¢ and '1/J be c= functions on an open subset 0 c R 2 satisfying ¢ 2 + 'lj; 2 < 1. Let (x, y, u, v) denote the standard global coordinate system in TO = 0 x R 2 . Let F := Ju2

+ v2 + ¢(x,y) u + 'lj;(x,y) v.

(12.13)

By (5.28), F is pointwise projectively related to the following spray (12.14)

The Riemann curvature of sprays is determined by geodesics. If two sprays are pointwise projectively related, then the Riemann curvatures are related by a simple equation. Lemma 12.1.2 Let G and G := G- 2PY be sprays on ann-manifold M. The Riemann curvatures are related by Ry = Ry

+ 2(y)

I+ Ty y,

(12.15)

where Ty E T;M with Ty(y) = -3(y). Hence, the Ricci scalars R = n~l Ric and R = n~l Ric are related by R(y) = R(y)

+ 2(y).

(12.16)

Proof By (8.14), we obtain the following identity in a local coordinate system

;: ;' uk ri R- ik = Rik + '~

+ Tk

Yi

(12.17)

where

p2- P,kyk 3(P;k- PP.k) Observe that

TkYk

= 3(P;kYk -

P 2)

(12.18)

+ 3.k·

(12.19)

+ 23 = -3.

It follows from (12.17) that

R=

R + - 1 - (n3 - 3) = R + 3. n-1

(12.20)

12.1. PROJECTIVELY RELATED SPRAYS

177

This completes the proof.

Q.E.D.

The equation (12.15) was established in [Ma3] for Finsler metrics . Recall that a spray G is isotropic if the Riemann curvature is in the following form Rv = R(y)I + (y y. From (12.15), we see that if a spray is pointwise projective to an isotropic spray, then it must be isotropic too. By (12.15)-(12.19), we immediately obtain the following Proposition 12.1.3 Let (M, G) be a spray space and jectively related spray. (a)

R= R

G :=

G- 2PY a pro-

if and only if P satisfies

(12.21) (b) Ric = Ric if and only if P satisfies

yk P;k = p2_

(12.22)

The special case of Proposition 12.1.3 has been proved in Example 9.3.1. Comparing (12 .21) with (2.41), we make the following Definition 12.1.4 Let (M, G) be a spray space. Assume that a function P on T M is on T M \ { 0} satisfying

coo

\;/).. > 0.

P(>.y) = >.P(y),

(a) P is called a Funk function if it satisfies the following system of PDEs

P;k = ppk· (b) Pis called a weak Funk function if it satisfies the following PDE

ykP;k

= p2 _

A natural question is whether or not there always exist non-trivial Funk functions on a spray space. The following proposition tells us that there are no non-trivial Funk functions on compact spray spaces. Proposition 12.1.5 Let P be a weak Funk function on a spray space (M , G). Then for any geodesic c(t) in (M, G) with c(O) = y,

P(c) =

P(y)

(12 .23)

1- P(y)t

In addition, if M is compact without boundary, then P

= 0.

178

CHAPTER 12. PROJECTIVE GEOMETRY

Proof Let P(t)

:=

P(c(t)) . We have P'(t) = P;k(C(t))ck(t),

where (ck (t)) denote the coordinates of c( t). Hence P 2 (t)- P'(t)

= 0. Q.E.D.

Solving the above ODE, we immediately obtain (12.23) .

For sprays G and G = G - 2PY, Ric = Ric if and only if P is a weak Funk function (Proposition 12.1.3) . Proposition 12.1.5 can be generalized to the following Proposition 12.1.6 Let G be a positively complete spray space. Assume that a spray G = G - 2PY satisfies

Ric 2: Ric,

then the projective factor P satisfies p

In addition, if G and

G are

:S

0.

G = G.

both reversible, then

Proof Assume that P(v) > 0. Let c be a geodesic in (M, G) with c(O) = v . It follows from (12.16) that P(t) := P(c(t)) satisfies P'(t)- P 2 (t)

= -2(c(t)) = R(c(t))- R(c(t)) 2: o.

Let

Po(t) Po(t) satisfies

:=

(12.24)

P(v)

1- P( v )t

P~(t)- P0 (t) 2 = 0.

(12.25)

To compare P(t) with P0 (t), we define

h(t) := e- fo'

[P(s)+Po(s)] ds {

P(t)- Po(t)} .

It follows from (12 .24) and (12.25) that

h'(t) = e-

I:

[P(s)+Po(s)] ds {

P'(t)- P(t) 2

-

P~(t) + Po(t)

Since h(O) = 0, we conclude that

P(t)- P0 (t) 2: 0,

t 2: 0.

2}

2: 0.

12.1. PROJECTIVELY RELATED SPRAYS Let t 0 := 1/ P(v)

179

> 0. Then P(c(t 0 )) = lim P(t) ?: lim P0 (t) = +oo. t~t;

t~t;

This contradicts our assumption. Thus P(v) ;::; 0. If both G and G are reversible, then P satisfies

P( -v)

= -P(v)

;::; 0.

Thus P(v) = 0.

Q.E.D.

Let Go := yi 8~, denote the canonical flat spray on Rn and G a reversible spray on Rn with Ric?: 0. By Proposition 12.1.6, we conclude that if G is complete, then it can not be pointwise projective to G 0 unless G = G 0 . However, there are many incomplete sprays on Rn with Ric ?: 0, which are pointwise projective to G 0 .

Example 12.1.2 Consider the following spray on Rn

G

(x,y)

= Go + 21 + lxl2 Y.

(12.26)

A simple computation yields that Ric > 0. Note that G is incomplete and

d

~~G.

Let G be a reversible spray on Rn with Ric ;::; 0. By Proposition 12.1.6, we conclude that G can not be pointwise projective to G 0 unless G = G 0 . However, there are many reversible sprays on a proper domain n C Rn with Ric;::; 0, which are pointwise projective to G 0 . Example 12.1.3 Consider the following spray on the unit ball Bn C Rn

G

=G

o

- 2 (x, y) Y 1 -lxl2 .

An easy computation yields Ric< 0. Note that Bn is a proper domain of Rn and G

~

G0.

(12.27)

CHAPTER 12. PROJECTIVE GEOMETRY

180

12.2

Projectively Related Finsler Metrics

A Finsler metric on an open subset in Rn is called a projective Finsler metric if it is pointwise projective to the Euclidean metric. The problem of characterizing and studying projective Finsler metrics is known as Hilbert's Fourth Problem. More general, given a Finsler metric on a manifold, we would like to determine all Finsler metrics on the manifold whose geodesics coincide with the geodesics of the given one as set points. A. Rapcsak [Rap] proved the following important results. Lemma 12.2.1 Let (M,L) be a Finsler space and La Finsler m etric on M . Then the spray coefficients (ji and Gi satisfy

(;i = Gi

+~flit { L;k·tYk- L;t },

(12.28)

where L;l and L;k·l are the covariant derivatives of L with respect to L given by

Proof Observe that

Q.E.D.

Then (12.28) follows immediately.

Theorem 12.2.2 (Rapcsak) Let (M, L) be a Finster space. A Finster metric L on M is pointwise projective to L if and only if -

k

k - _ L;kY L.uy - L.t - ----L.t. 2L ' ' In this case, the spray coefficients are related by (;i = Gi

+ Pyi,

where

k

-

p

(12.29)

'---. = -L.ky 4L

(12.30)

Proof Suppose that L is pointwise projective to L. By Lemma 12.1.1, the induced sprays G = yi 8~, - 2Gi (y) 8~, and G = yi 8~, - 2Gi(y) 8~, satisfy

(12.31)

where P is a positively homogeneous function on T M \ {0}. By (12.28), we obtain i k (12.32) L;k·tY - L;t = 4P 9aY = 2P L .t. It follows from the homogeneity of L;k that -

l

-

L ;k·lY = 2L;k·

12.2. PROJECTIVELY RELATED FINSLER METRICS This implies

kl (L;k·lY - L;l)y

-k = 2L;kY -

-l L ;lY

181

-k = L;kY .

Contracting (12.32) with y 1 yields -

k-

l

-l

-

(L ;k·lY - L;l)y = 2P L.1y = 4P L. This gives (12.30) . Plugging (12.30) into (12.32) gives (12.29). The converse is trivial, so is omitted. Q.E.D. Corollary 12.2.3 Let L = L(x , y) be a Finsler metric on an open subset U C Rn. Then L is pointwise projective to the standard Euclidean metric if and only if

(12.33) -

k

In this case, the spray coefficients C;i of L are given by C;i = L~L yi . Let L and L be Finsler metrics on an n-dimensional manifold M. Assume that L is pointwise projective to L. Then L and L satisfy (12 .29). The spray coefficients of Land L are related by C;i = Gi + Pyi , where Pis given by (12 .30) . Recall (12 .15) and (12 .16)

Rv

=

R(y)

Ry + 3(y)I + Ty y,

(12 .34)

R(y) + 3(y),

(12.35)

where 3(y):

p2- P;lYl,

ry(u):

3(P;k- PP.k)uk + '2 .kuk.

(12.36) (12.37)

Let A(y) := R(y)j L(y) and 5.(y) := R(y)/ L(y). (12.35) is equivalent to the following equation

5.(y)L(y)- A(y)L(y)

-

= 3(y) = [L;k~ 4L

k

]z-

-

[L;k~ 4L

k

] yl. ;l

(12.38)

Theorem 12.2.4 ([Sz2](MaWe]) Let (M, L) be ann-dimensional Finsler space and L another Finsler metric pointwise projective to L. Suppose that L is of scalar curvature A(y). Then L is of scalar curvature 5.(y) which is given by (12.38).

Proof By assumption, L is of scalar curvature A(y) , namely, the Riemann curvature is in the form Ry

= ,\(y)L(y)I + (y y.

(12.39)

CHAPTER 12. PROJECTIVE GEOMETRY

182

Plugging (12.39) into (12.34) yields

Ry where 5.(y)L(y) := >.(y)L(y)

= 5.(y)L(y)I + (y y,

+ 2(y ) and (y

:= (y

(12.40)

+ Ty·

Q.E.D.

For a Riemannian metric g on a manifold of dimension n 2: 3, it is of scalar curvature if and only if it is of constant curvature. If g and g both are pointwise projectively related Riemannian metrics, then in dimension n 2: 3, g is of constant curvature if and only if g is of constant curvature. The same statement is also true for Einstein metrics. More precisely, we have Theorem 12.2.5 (J. Mikes [Mik1][Mik2]) Let (M, g) be ann-dimensional Riemannian space and g another Riemannian metric pointwise projective to g. Suppose that g is Einsteinian, then g must be Einsteinian.

Now we consider positive definite Finsler metrics. As usual, we denote a positive definite Finsler metric L by its square root, F = VI. Let (M, F) be a positive definite Finsler space and F another positive definite Finsler metric on M. Plugging L = F 2 in (12.28) yields (12.41) where p

(12.42)

(12.43) Here the covariant derivatives ofF are taken with respect to F. We immediately obtain the following Theorem 12.2.6 (Rapcsak) For two positive definite Finster metrics F and F on a manifold M , F is pointwise projectively related to F if and only if F satisfies k (12.44) F;k·lY - F;l = 0. In this case, the spray coefficients are related by C;i

= Gi + Pyi,

where

k

-

_ F';kY P -----.

2F

Example 12.2.1 Let (3 = bi(x)yi be a 1-form and F = F(y) a positive definite Finsler metric on a manifold M. Consider F :=F+(3.

12.2. PROJECTIVELY RELATED FINSLER METRICS

183

Adding a 1-form to a Finsler metric is called a Randers change. Suppose that 11/JII := supF(y)=l i,B(y)l < 1. F is still a positive definite Finsler metric. Observe that /3 ;l

Bb; = ( Bxl k

-

8b1 s ) k Bxk - f 1kbs y .

(

f3;k·lY =

rsil bs) y i '

Thus By Lemma 4.2.2, We obtain

k F;k ·lY - F;l

= /3;k·lY k -

8b1 = ( Bxk

/3;!

Bbk) k - Bxl y .

Thus (12.44) holds for F = F + f3 if and only if f3 is close. That is, F = F + f3 is pointwise projective to F if and only if f3 is close. This result is obtained by Hashiquchi-Ichijy6 [Halc3] (see also Example 3.3.1.1 in [AnlnMa]) . U Now we derive several conditions equivalent to (12.44). Let F be a positive definite Finsler metric on a positive definite Finsler space (M, F). Let

p Observe that

-

[PF].l

Thus

F.kyk

:= - '---.

2F

1k 1= -Fuy + -F1. 2 ' 2 '

F satisfies (12.44) if and only if

fl;1

=

[PFk

(12.45)

F satisfies

(12.45) for some P satisfying P(>..y) = >..P(y), V>.. > 0. By contracting (12.45) with y 1 , we obtain P = ~ 1; 1 • This proves the following

Suppose that

Proposition 12.2. 7 Let (M, F) be a positive definite Finster space and

F

another positive definite Finsler metric on M. F is pointwise projective to F if and only if there is a positively homogeneous function P on T M, i.e., P(>..y) = >..P(y), V>.. > 0, such that F satisfies (12.46)

In this case,

-

k

_ --F';kY . p_ 2F

and the spray coefficients are related by {;i = Gi

+ Pyi.

CHAPTER 12. PROJECTIVE GEOMETRY

184

By Propositions 12.1.3 and 12.2.7, we immediately obtain the following Proposition 12.2.8 Let (M, F) be a positive definite space and F a positive definite Finster metric on M . Assume that F satisfies

(12.4 7) for some positively homogeneous function P on T M . (a)

R= R

if and only if P is a Funk function on (M, F);

(b) Ric= Ric if and only if P is a weak Funk function on (M, F).

A natural question arises: Given a (weak) Funk function P on a positive definite Finsler space (M, F), whether or not there exists a positive definite Finsler metric F on M satisfying (12.47)? This inverse problem has not been studied yet. Let us take a look at the special case when P is a Funk function on an open domain D c 1Rn . By Proposition 12.1.3, we know that for any Funk function P on TD, the spray G := yi 8~, - 2P 8~. is R-flat. By [GrMu],

G is locally induced by a (possibly not positive definite) Finsler metric F on

n.

According to Proposition 12.2.7, F is a solution of (12.47). Therefore there exists non-trivial pointwise projectively flat R-flat Finsler metrics. This is just Theorem 10.3.5. See further discussion in Section 12.4 on the inverse problems.

We continue to derive conditions equivalent to (12.44) or (12.47). Let F be a positive definite Finsler metric on a positive definite Finsler space (M, F). Assume F satisfies (12.47) for some positively homogeneous function P on TM . This implies (12.48) Contracting (12.48) with yi gives i_FF- ;i·iY i_F;j·iY ;j ·

We obtain another equivalent version of Rapcsak's Theorem. Proposition 12.2.9 Let (M, F) be a Finster space and F another Finster metric on M. F is pointwise projective to F if and only ifF satisfies

(12.49) We still assume that F is pointwise projective to F. As a scalar function on T M, the covariant derivatives ofF with respect to F satisfy the following Ricci identities (12.50)

F;k·l = P.l;k, F.;.j;k = P .i;k·i

+ FmBiJk·

(12.51)

12.2. PROJECTIVELY RELATED FINSLER METRICS

185

Here Bf]k denote the Berwald curvature coefficients. Differentiating (12 .44) with respect to yi yields (12.52) With both (12.50) and (12.48), we obtain (12.53) It follows from (12.51) and (12.53) that

(12.54) Thus (12.54) is a necessary condition for F to be pointwise projective to F. This necessary condition was obtained by M. Matsumoto [Ma9] in 1993. Corollary 12.2.10 ([Ham]) Let F = F(x, y) be a positive definite Finster metric on an open subset U C lR.n. Then F is pointwise projective to the standard Euclidean metric if and only if one of the following conditions is satisfied. -

F x kyt

k

-

= Fxt,

(12.55)

= Fxtyk .

(12.56)

F x kytY

Any of the above conditions implies the following

(12.57) If F 1 , F 2 are two pointwise projectively related positive definite Finsler met-

rics and a 1 , a 2 two positive numbers , then

also satisfies (12.44) . Thus F is pointwise projective to F 1 and F2 , provided that F is still a positive definite Finsler metric. A positive definite Finsler metric F on an open subset U C Rn is called a projective Finster metric if it is pointwise projective to the Euclidean metric. Projective Finsler metrics are, of course, pointwise projectively flat. Conversely, every locally projectively flat Finsler metric is locally isometric to a projective Finsler metric. There are lots of projective Finsler metrics. Two dimensional projective Finsler metrics are studied in [Bw5][Ham][Ma10][Mall]. In [Bw5], L. Berwald gives explicit formulas for all two-dimensional projectively flat Finsler metrics with constant main scalar. See [Alv1][Alv2][AlGeSm] for some constructions in higher dimensions. Let F denote the Funk metric on a strongly convex domain Lemma 2.3.1, F satisfies

n in

Rn. By

(12.58)

186

CHAPTER 12. PROJECTIVE GEOMETRY

It follows from (12.58) that

FxkylYk

= Fx1·

By Corollary 12.2.10, we conclude that F is pointwise projective to the Euclidean metric. More precisely, the spray coefficients ofF are given by Gi(y) P(y)yi where

Thus

::;' - p2 - p xk yk -- - ~ 4 F2 .

~-

By Theorem 12.2.4, F is of constant curvature K = -1/4. Let F(y) := F( -y) denote the reverse of F. By the same argument, one can easily show that Pis also pointwise projective to the Euclidean metric. Further, P is also of constant curvature K = Since both P and F are pointwise projective to the Euclidean metric, so is P := ~(P +F). The spray coefficients (ji = f>y i of Pis given by

i-.

p=

Fxkyk = ~(F- F) . 2F 2

Observe that

This gives :.=::

=

p2- P;kYk

=

_fr2 .

By Theorem 12.2.4, we conclude that the Klein metric

P has constant curvature

K = -1. We summarize as follow.

Theorem 12.2.11 ([Fu1][Bw3][0k]) Let n be a strongly convex domain in Rn. Let F denote the Funk metric on n, P the reverse ofF and P := ~(P +F) the Klein metric on D. F, F and F are pointwise projective to the Euclidean metric on R n. Both F and P have constant curvature K = -1/4, while P has constant curvature K = -1.

12.3

Projectively Related Einstein Metrics

In this section we will continue to study projectively related Finsler metrics. We will show that if two Finsler metrics are pointwise projectively related, then along each geodesic, their Ricci scalars are related by a simple equation. This leads to some important results on projectively related Einstein metrics. Let F and P be positive definite Finsler me tries on a manifold M . Put Ric(y) = (n- l)>.(y) F 2 (y),

Ric(y)

= (n- 1)_\(y) F2 (y),

187

12.3. PROJECTIVELY RELATED EINSTEIN METRICS where >{y) and 5. (y) are scalar functions on T M. Suppose that F and pointwise projectively related. Then (12.38) holds, i.e.,

F are

(12.59) Let c(t) be a unit speed geodesic ofF and c(i) be a unit speed geodesic of such that c = c as a set of points, namely, c(t) = c(i) , Let

P

di - >0. dt

A(t)

:=

A(c(t)),

5.(i)

f(t)

:=

1 [PW'

- f(t) =

:=

5.(i:(i)). 1

JF(J)"

It follows from (12 .59) that

f"(t)

+ A(t)f(t) =

]"(i)

+ 5.(i)](i)

=

~~;).

(12.60)

~(t~

(12.61)

J3(t)

.

Equations (12.60) and (12.61) can also be expressed as follows.

-~(di)d 3 i

~(d 2 t)2 A(t)(di)2 = dt + 4 dt 2 +

5.(i)(dt)4. dt

(12.62)

-~(d:)d~t +~(d~t)2 +5.(i)(d:)2 =A(t)(d:)4.

(12.63)

2 dt dt 3

4 dt 2

dt dt Note that (12.63) is the inverse of (12.62) and vice versa. 2 dt dt 3

From now on, we assume that F and F are Einstein metrics, i.e., A and 5. are constants. In this case, (12 .60) is solvable, namely, the general solution can be expressed in terms of elementary functions. The solution of (12.60) with

f(O) =a> 0, is determined by

1

f(t)

a

/'(0) = b ¥: 0

s

----r=====:::::;:ds = ±t, J- As4 + 2Cs 2 - 5.

where the sign ± in (12.64) is same as that off' (0)

= b and

(12.64)

CHAPTER 12. PROJECTIVE GEOMETRY

188

When b = 0, the solution can be obtained by letting b -+ 0. Note that (12 .65) and (12.66) Thus the integrand in (12 .64) is defined for s close to a and the maximal solution f(t) > 0 is defined on an interval I containing s = 0. By evaluating the integral in (12 .64), we can prove the following Theorem 12.3.1 ([Sh7]) Let F and F be pointwise projectively related positive definite Finsler metrics on a compact n-manifold M. Suppose that both F and F are Einstein metrics with

where A). E { -1, 0, 1}. Then A and ); have the same sign. More details are given below. (i) If A = 1 = .>;, then along any unit speed geodesic c(t) ofF

F(c(t)) =

2 ( a2

-

1/a2

-

b2 )

cos(2t)

+ 2absin(2t) + (a 2 + 1/a2 + b2 )

,

(12.67) where a > 0 and -oo < b < oo are constants. Thus, for any unit speed geodesic segment c of F with length of 1f, it is also a geodesic segment of F with length of 1f . (ii) If A = 0 = 5;, then along any geodesic c(t) ofF or

F,

F(c(t)) _ =constant . F(c(t)) (iii) If A= -1

= 5;,

then

(12.68)

F =F.

To prove Theorem 12 .3.1 , we divide the problem into several cases.

Case 1: A = 1. From (12.64), we obtain f(t)

= J(a 2 -

C) cos(2t)

+ absin(2t) +C.

(12.69)

Using (12 .65) we can rewrite (12.69) in the following form f(t)

=

Jcz-

.>;sin [sin- 1

(

aZ-

c_) ± 2t] + c,

Jcz -A

(12 .70)

12.3. PROJECTIVELY RELATED EINSTEIN METRICS

189

where the sign ± in (12. 70) is same as that off' (0) = b when b =j:. 0. Otherwise, the sign can be chosen arbitrarily. Case 2 : A = 0. From (12.64) we obtain f (t) =

V+ (a

bt)

2

+ :\ ( ~) 2 .

(12. 71)

+ absinh(2t) - C .

(12.72)

Case 3: A= -1. From (12.64) , we obtain f(t) = J(a 2 +C) cosh(2t)

Using (12.65) we can rewrite (12.72) as follows

f(t)

=

Jvc Jv

2

+ ;\

-C2

cosh [ cosh- 1

(~) ± 2t]

- C

if C 2

Je±2t(a2 +c)- C -

;\

sinh [ sinh- 1

+ ;\ > 0

if C 2 + ;\

(m) ± 2t] -

The sign ± in (12.73) is same as that of f' (0) = b =j:. 0.

C

if C 2

=0

+ ;\ < 0 (12. 73)

The proof of Theorem 12.3.1 follows from the above formulas (12.69)-(12. 73) for f(t) = [F (c(t) )]- 1 12 . First, we know that any Finsler metric on a closed manifold is complete. Thus f(t) > 0 must be defined on ( -oo, oo). The completeness ofF implies

0 /_ 00

!(~)2 dt = 00 =

1 !(~)2 00

dt .

Then the Einstein constants must have the same sign. The reader can easily Q.E.D. prove Theorem 12.3.1 (i)- (iii). The non-compact case is more complicated. We list some theorems below. Theorem 12.3.2 ([Sh7]) Let F and P be pointwise projectively related positive definite Finsler metrics on a non-compact n-manifold M. Suppose that both F and F are Ricci-fiat. Then along any unit speed geodesic c(t) ofF, F(c(t))

=

1 (a+bt)2'

(12. 74)

where a > 0 and -oo < b < oo are constants. Thus F is complete if and only if In this case, along any geodesic c(t) ofF or P,

P is complete.

F( .(t)) = constant . _ c F(c(t))

190

CHAPTER 12. PROJECTIVE GEOMETRY

We actually prove that if c is a unit speed geodesic ofF, along which F / P ::/= constant, then c can not be defined on ( -oo, oo) . If c is defined on [0, oo) or ( -oo , 0], then it has finite F-length. This suggests that there might be a positively complete (Ricci-)flat metric and a negatively complete (Ricci-)flat metric which are pointwise projective to each other. Such examples have not been found yet. Theorem 12.3.3 ([Sh7]) Let F and P be pointwise projectively related positive definite Finster metrics on a non-compact n-manifold M. Suppose that F and P are Einstein metrics with

Ric

= - (n -

1) F 2 ,

Ric

= - (n -

1)

F2 .

Then along any geodesic c(t) ofF,

-

F(c(t)) = (

- 1/a2

+ a 2 + b2

where a > 0 and -oo then F = P.

)

2 cosh(2t)

+ 2absinh(2t)-

(

- 1/a2

-

a2

) ,

+ b2

(12.75)

< b < oo are constants. If both F and P are complete,

Corollary 12.3.4 Let F be a Finster metric on a strongly convex domain Rn. Suppose that the following hold

n in

(a) F is complete; (b) the geodesics ofF are straight lines; (c) F has constant curvature Then F is the Klein metric on

12.4

>.

= -1.

n.

Inverse Problems

A spray G on a manifold M is said to be locally Finslerian if at any point x E M, there is a neighborhood Ux in which G is induced by a Finsler metric F. G is said to be globally Finslerian, if it is induced by a Finsler metric on the whole manifold. Problem 1: Given a spray, is it locally Finslerian? If it is locally Finslerian, is it globally Finslerian ?

12.4. INVERSE PROBLEMS

191

If a two-dimensional spray G is induced by a Finsler metric L, then the Riemann curvature takes the following form (8.62), namely,

Ry(u) = K(y){ L(y)u- gy(y, u)

y}

where K(y) = Ric(y)/ L(y) . Thus R = 0 if and only if Ric = 0. This implies that any two-dimensional spray G with Ric = 0 and R =J 0 is not Finslerian. A spray on a manifold is said to be locally projectively Finslerian if at each point, there is a neighborhood in which it has the same geodesics as a Finsler metric. A spray is said to be globally projectively Finslerian if it has the same geodesics as a Finsler metric on the whole manifold. The following is the famous Hilbert's Fourth Problem.

Hilbert's Fourth Problem: Given a domain D c Rn, determine all Finster metrics on n whose geodesics are straight lines. There are a number of papers on the Hilbert Fourth Problem. See [Alv2] [Ale] [Dar] [Po] [Sz3] [Za],etc. In Hilbert 's Fourth Problem, the base metric is the standard Euclidean metric on Rn . Thus if a Finsler metric is pointwise projective to the Euclidean metric, then it must be of scalar curvature (Theorem 12.2.4). It turns out that there are many Finsler metrics on a domain in Rn which are pointwise projective to the Euclidean metric. In particular, the Funk metrics and the Klein metrics are all pointwise projective to the Euclidean. Some Randers metrics are also pointwise projective to the Euclidean metric. For a Riemannian metric a (y) = g (y, y) and 1- form f3 on a domain D C R n, if a is pointwise projective to the Euclidean metric and d/3 = 0, then the Randers metric F(y) := a(y) + f3(y) is also pointwise projective to the Euclidean metric. The following problem is natural.

J

Problem 2: Given a spray, is it locally projectively Finslerian ? If it is locally projectively Finslerian, is it globally projectively Finslerian ? Before we discuss Problem 2, let us take a look at the following

Example 12.4.1 Consider the following spray on R 2 G

where r

fJ fJ VJ fJ UJ fJ = ufJx - + v - - - u 2 + v2 - +- u 2 + v2 - , fJy r fJu r fJv

(12. 76)

> 0. By Example 4.1.3, we know that the geodesics of G are given by x(t)

r cos

(~t +B)

y(t)

rsin

(~t +e)- rsinB + y(O).

- r cosB + x(O),

CHAPTER 12. PROJECTIVE GEOMETRY

192

Thus the geodesics of G are circles of radius r. We are going to show that G is locally pointwise projective to a Randers metric everywhere. For any point (xo, Yo) E R 2 , define

F

vu

:=

2

k

1-k Xo) v, + v2 + - -r ( y - Yo) u- -(xr

(12.77)

where k is an arbitrary constant. F is indeed a Randers metric on an open domain n in R 2 , where n is given by

(~f(x-xo)2+ c~kf(Y-Yo)2 < 1. When k = 0, 1, n is an open subset between two lines. When k enclosed by an ellipse.

I

\

I

.

I

I

\I

\

I

\I

I

,'

....

\

n is

\ \

.. . .

V

W'

0, 1,

... --- .

\1

\1

=f.

l

,,~~-t-:~~:--- ,~--t--,--- -,>,--t--,~---- ,~ -~-~ II

I

\

\

,

\I

I

.,

'

fi

.,..,'

\\

'~-,'

\I \

\I

II

,

\I

'......

,'

/!

\~1

\I \

'II

' .......

..... ___ ...

... ___ ...

k

= 0.25, r = 1,

(x 0 , Yo)

= (0, 0)

To prove that F is pointwise projective to G on spray ofF,

-a au

a ay

a ax

-

n,

we need to compute the

-a av

G =u- +v- -2G- -2H-.

According to Example 5.2.2, the spray coefficients

6

and

fi

ofF are given by

+ v2

6

Pu + 3:__Ju 2

ii

Pv- ..!:!:_ 1u 2 + v2 ' 2r v

2r

where 1- 2k

1 { P = 2F - r - uv-

vu2

+ v2 ( k(x- X )u + (1- k)(y- Yo)v ) } .

r2

0

Thus the spray G is pointwise projective toG on n. Note that when k changes, the domain n of F changes too. But the orbits of the geodesics remain unchanged.

12.4. INVERSE PROBLEMS

193

We will show that the spray G is not globally projectively Finslerian. More precisely, it is not pointwise projective to any positive definite Finsler metric. See Example 14.1.1 below. U Therefore, another problem arises: If a spray on a manifold is (locally) Finslerian, is it induced (locally) by a positive definite Finsler metric ? In [Sz1] , Z.I. Szab6 studied the following inverse problem on affine sprays. Problem 3 : Given an affine spray, under what sufficient and necessary conditions, the spray is locally Finslerian ? By definition, a spray is affine if the Berwald connection is an affine connection. A Finsler metric is Berwaldian if the induced spray is affine. Z. Szabo first proved that if an affine spray is induced by a positive definite Finsler metric, then it can be induced by a Riemannian metric. Based on this observation, he proved the following Theorem 12.4.1 (Szab6) Let G be an affine spray on a manifold M whose Riemann curvature R does not vanish everywhere. Then G is induced by a non-Riemannian positive definite Berwald metric if and only if the following two conditions are satisfied: (a) G is induced by a Riemannian metric. (b) The Berwald connection of G is either locally reducible or it is a locally irreducible, locally symmetric connection of rank k 2: 2.

We remark that for a given affine spray on a manifold, it is hard to check whether or not the condition (a) in Theorem 12.4.1 is satisfied. Nevertheless, by Theorem 12.4.1, Szabo concludes that a positive definite Finsler metric inducing an affine spray must be one of the following four types: (a) F is Riemannian; (b) F is locally Minkowskian ; (c) F is non-Riemannian and the Berwald connection Dis locally irreducible, locally symmetric of rank r 2: 2; (d) F is non-Riemannian and the Berwald connection D is locally reducible. In this case, the space can be locally decomposed to a Descartes product of Riemannian space, locally Minkowskian spaces and locally irreducible, locally symmetric non-Riemannian positive definite Berwald spaces of rank r 2: 2.

CHAPTER 12. PROJECTIVE GEOMETRY

194

Szabo gave a full list of 56 kinds of irreducible and globally symmetric Berwald metrics. As we know, affine sprays and isotropic sprays are two important classes of sprays. Thus it is natural to investigate the inverse problem on isotropic sprays.

Problem 4: Given an isotropic spray, is it locally Finslerian '? If it is locally Finslerian, is it globally Finslerian '? The inverse problem on isotropic (semi)sprays has been studied by J. Grifone and Z. Muzsnay [GrMu] . The results of their study can be applied to construct Lagrange metrics and Finsler metrics of scalar curvature. Now we make few remarks on the inverse problems on semisprays. By definition, a semispray s on u = (a, b) X 0 is locally Lagrangian if for any (s' 'TJ) E u' there exists an subinterval (a', b') of (a , b) and an open neighborhood 0' of 'TJ such that S on U' = (a' , b') x 0' is induced by a Lagrange metric on U'. See Chapter 3. We may ask the following questions.

Problem 5: Given a semispray, is it locally Lagrangian '? If it is locally Lagrangian, is it globally Lagrangian '? Problems 2 is closely related to Problem 5.

Proposition 12.4.2 Let U = (a, b) x 0 C Rn be an open subset. Given a semispray S = gs +~a &~a +q.a(s , ry, ~)&~a and a spray G = yi 8~, - 2Gi(x, y) 8~, on U. Suppose that they are related by

a= 2,···,n.

(12. 78)

S is locally induced by a Lagrange metric ¢> = ¢>( s , ry, ~) on U if and only if G is locally projectively induced by a Finster metric L = L(x, y) on U in the form (12. 79)

where s

= x 1 , rya = xa

and ~a= yajy 1 , a= 2, · · · ,n.

Proof We first assume that S is locally induced by a Lagrange metric ¢; on U. Define a Finsler metric L by (12 .79) . By Proposition 3.3.1, the spray G = yi 8~. - 2Gi (x, y) 8~, of L is related to the semispray S by a= 2, · ··,n.

(12.80)

It follows from (12.78) and (12.80) that there is a function P(x, y) such that (;i(x,y) = Gi(x,y) + P(x,y)yi. Thus G is locally pointwise projective to G. Namely, G is locally projectively induced by L.

12.4. INVERSE PROBLEMS

195

Conversely, we assume that G is locally projectively induced by a Finsler metric L in the form (12.79) for some function ¢ (which must be a Lagrange metric) . We claim that¢ induces the semispray S. Let G = yi 8~, -2Chx, y) 8~,

ts

denote the spray of L and S = +~a &~a + ~a(s, ry, ~)&~a the semispray of¢. By Proposition 3.3.1, G and S are related by

a= 2, · ·· ,n .

(12.81)

On the other hand , by assumption, G is pointwise projective to G. Namely, there is a function P(x, y) such that Gi(x, y) = Gi(x, y)+P(x, y)yi . This implies that

Notice that the right hand side of (12.82) is equal to a(s, ry, ~)by (12.78). Thus S is the semispray of¢. Q.E.D .

~a = q>a . This implies that S =

Theorem 12.4.3 (G . Darboux [Dar)) Every two-dimensional spray is locally pmjectively Finslerian . Every one-dimensional semispray is locally Lagrangian. G. Darboux [Dar] proved that for any equation (12.83) there is a Lagrangian such that its Euler-Lagrange equation is (12.83). One can easily construct a (singular) Finsler metric in an open domain in R 2 such that its geodesics satisfy (12.83). Thus every two-dimensional spray is locally projectively Finslerian. M. Matsumoto provided a detailed formula of Finsler metrics for certain types of equations (see [Mall)). In higher dimensions, D.R. Davis [Davl]-[Dav3] and D. Kosambi [Ko1][Ko2] first studied Problem 5. Finally, J. Douglas [Dg2] gave a complete solution to Problem 5 on semisprays in dimension two (i.e., semisprays on (a, b) X n, where dim n = 2). In [CSMBP], the authors gave a geometrical interpretation of the key parts of Douglas's paper [Dg2]. In [SCM], the authors made a general analysis on the integrability for the Helmholtz equations . They proved that a particular class of systems with n degree of freedom, corresponding to Class I of Douglas's classification in the two degrees offreedom case, is always Lagrangian . In [CPST], the authors extended other Douglas's results. In [GrMu], Grifone and Muzsnay solved the inverse problem for isotropic semisprays. This kinds of inverse problems form an important subject in mathematics.

Chapter 13

Douglas Curvature and Weyl Curvature There are two important projective invariants of sprays and Finsler metrics. One is a non-Riemannian projective invariant constructed from the Berwald curvature. The other is a Riemannian projective invariant constructed from the Riemann curvature. In this chapter, we will discuss these two projective invariants.

13.1

Douglas Curvature of Sprays

Let G = yi 8~, - 2Gi(y) 8~, be a spray on a manifold M. Recall that for any vector y E TxM \ {0}, the Berwald curvature By: TxM x TxM X TxM-+ TxM is a trilinear form By(u,v,w) = B)k 1(y)uivkwl 8~, lx defined by (13.1) The mean Berwald curvature Ey : T xM xTxM-+ R is a bilinear form Ey(u, v) = Eij(y)uivJ defined by (13.2) Based on the Berwald curvature, J. Douglas [Dg1] introduced a new quantity Dy : TxM x TxM x TxM -+ TxM which is a trilinear form Dy(u, v, w) = · k t a · Djk 1(y)u 3 v w ox' lx defined by (13.3) We call D :=

{Dy}yETM\{O}

the Douglas curvature. See also [Bw8].

197

198

CHAPTER 13. DOUGLAS CURVATURE AND WEYL CURVATURE

Let IIi = Gi- _1_Nm(y)yi.

n+ 1

m

II = yi a~· - 2IIi(y) a~· is the local projective spray associated with G. II depends only on the pointwise projective class of G. Thus any (local) invariant defined by II is a (local) projective invariant. Observe that (13.4) (13.5) Thus D is a projective invariant. It follows from (13.5) that Dy is a symmetric trilinear form. Moreover,

Dy(y,v,w)

= 0,

trDy(v, w) = 0.

(13.6)

Here trDy(v, w) := ~D]km (y)viwk denotes the trace of Dy. The following lemma is trivial. Lemma 13.1.1 E = 0 if and only if D =B . Proof Suppose that E = 0. It follows (13 .3) that D = B. Suppose that D =B. By (13.6), we obtain

Thus E = 0.

Q.E.D.

Proposition 13.1.2 Every spray G on a manifold M is globally pointwise projective to a spray G with B = D (equivalently, E = 0) . Hence G is globally pointwise projective to an affine spray G if and only if D = 0. Proof Take an arbitrary volume form dJ.L on M. Let S denote the S-curvature of (G, dJ.L). Define 2S G:=G+--Y. (13.7) n+1

By Proposition 5.2.1, we know that S = 0. Hence from Lemma 13.1.1 that B = f> =D.

E=

0 by (6.13). It follows Q.E.D.

Proposition 13.1.2 is a strength of the Douglas theorem ((Dg1][Bw8]). Douglas only proved a local version.

13.2. PROJECTIVELY AFFINE SPRAYS AND DOUGLAS METRICS 199

Projectively Affine Sprays and Douglas Met-

13.2

.

riCS

Proposition 13.1.2 suggests the following Definition 13.2.1 ([Dg1]) A spray is projectively affine if D = 0. A Finsler metric is called a Douglas metric if its spray is projectively affine.

We are going to study projectively affine sprays and Douglas metrics. First of all, let us take a look at the following Example 13.2.1 Consider a generalized Randers metric F = F + f3 on a manifold M, where F = F(y) is a Finsler metric on M and f3 = bi(x)yi is a 1-form on M such that supF(y)=l l/3(y)l < 1. By (12.41), the spray coefficients ofF and F are related by (13.8) -

Fky

k

where P = -'---and 2F (13.9) Since F,k = 0, we obtain k F;k·tY - F;t

This gives

= f3;k·tY k -

f3;t

= (bt;k

k - bk;t)Y .

k p l 2:fl (bt;k- bk;t)Y . Thus if /3 is close, i.e., br;s- bs;r = 0, then D =D. In this case, F

.

Q' =

is a Douglas metric if and only ifF is a Douglas metric. This generalizes a result of S. Bacs6 and M. Matsumoto [BaMa2]. They proved that for a Randers metric F = a+/3, ~ F is a Douglas metric if and only if f3 is a close 1-form.

One of the important problems in Finsler geometry is to describe Landsberg metrics which are not Berwaldian. So far, no explicit examples have been found yet. The following result shows that such examples do not exist among Douglas metrics. Proposition 13.2.2 ([Bai1Ki][BaMa1]) For a Douglas metric L on a manifold M, if L = 0, then B = 0. Proof We will sketch a proof for a positive definite Douglas metric F(y) ·-

yY(ij).

It follows from (13.3) that

(13.10)

200

CHAPTER 13. DOUGLAS CURVATURE AND WEYL CURVATURE

where Eij := ~Bi'Jm· Let 1

k

hij := 9ii - F 2 9ikY 9itY

l

and h):= gikhjk = oj- -hgjly1yi. Assume that F is also a Landsberg metric, i.e., Lijk = -~Y 8 9stBLk Contracting (13.10) with h/ yields

= 0.

(13.11) Substituting (13.11) into (10.12) yields Eikhjt

+ Eikhil- Eilhjk

- Ejthik

= 0.

(13 .12)

Contracting (13.12) with gik yields, p E·t= --h·t 1 n- 1 1 '

(13.13)

where p := gii Eij. Plugging (13.13) into (13.11) gives (13.14) Plugging (13.13) into (13.10) gives

n 22 ~ 1 { hikhj + hjkhi + hijh'k} + (n 2

~ 1)F { a~k (Fp)hii + 2pFCijk }ym.

(13.15)

Comparing (13.15) with (13.14), we obtain 8(pF)

hii~

= -2pFCiik·

(13.16)

Contracting (13.16) with giJ yields

(n-

8(pF)

1)~

= -2pFCk,

(13.17)

where Ck := giiCiik· Substituting (13.17) into (13.16), we obtain (13.18) Assume n > 2. Then (13.18) implies that either Ck = 0 or p = 0. If Ck = 0, then F is Riemannian by Deike's theorem [De][Bk2] . If p = 0, then E = 0 by (13.13). This implies B = 0 by (13.10). In either case, F is always a Berwald metric.

13.2. PROJECTIVELY AFFINE SPRAYS AND DOUGLAS METRICS 201 This fact is due to S. Bacs6 and his collaborators [BallKi] [BaMa1] (claimed by H . Izumi [Iz] in 1984) . In [BallKi], the Douglas metric is assumed to be positive definite as above. Later on, this condition was removed in [BaMa1] . The proof in dimension two is omitted. Q.E.D. From Proposition 13.2.2, we immediately obtain the following

Corollary 13.2.3 Let F be a positive definite R-quadratic Finsler metric on a compact manifold M. F is a Douglas metric if and only if it is a Berwald metric. Proof Assume that F is a Douglas metric. By Theorem 10.3.2, F must be a Landsberg metric. Thus F must be a Berwald metric by Proposition 13.2.2. Q.E.D.

It is much easier to describe projectively affine sprays than Douglas metrics. Let G = yi a~ · - 2Gi(x,y)a~· be a spray on an open subset u =I X n c Rn . Let

a= 2,··· ,n,

G1 :=

0,

0 a ·= .

1

--ylyl..t, y) = ¢(t, >..y),

'.. > 0.

(14.2)

For simplicity, we set i 8 I Y -Y ~ x· ux'

(14.2) gives rise to (14.3) By the mean value theorem, there is 0

< t>-. < >.. such that

14.1. EXPONENTIAL MAP OF SPRAYS as A-to+. Thus

f) Jk

.

-8 . (x',O)y 1 yJ

.

223

= y k.

This implies that d(expx)lo =identity at the origin 0 E TxM. Differentiating (14.3) with respect to yi yields [)jk

i

i

fJyi ((x ),A(y ))

Note that

f)q}

1 f)q}

= :\ fJyi (A,y) .

-8 . (O,y) yJ

(14.4)

= 0.

We obtain f) Jk

( . .) fJyi (x'), A(y')

Thus

. i ) ) f) ( k k hm -f)8f k. ( (x i ),A(y) = -8f) . ( f)q} f)\ (O,y) = -8 . y) =bj . yJ yJ /\ yJ

.X-tO+

That is,

f

is C 1 along the set of zero sections of T M.

Q.E.D .

By Theorem 14.1.1, we immediately obtain the following

Corollary 14.1.2 Let (M, G) be a spray space. For any point x E M, there is a neighborhood Ux of x and a positive number rx > 0 such that for any x E Ux, expz : TzM -t M is a diffeomorphism on Bz(r) C TzM onto its image for all 0 < r < r x, where Bz(r)

:=

a Iz E TzM, ~ · 2< r 2} . {y'·fJxi ~(y') i=l

Now we study the regularity of exp on zero sections in T M. First, we assume that G is an affine spray, i.e., Gi(y) are quadratic functions in y E TxM for all x E M. By the O.D.E. theory, the solution cy to (14.1) smoothly depends on the initial data. Thus exp(y) = Cy(1) is c= on U. In particular, expx is C 2 at 0 E TxM for all x EM. Conversely, we assume that expx is C 2 at 0 E TxM for all x. By definition, for any r > 0 and compact subset U C M, there is a number E > 0 such that cy(t) = exp(ty), 0 S t S 1 + E, is a geodesic for every y E TU with F(y) < r. Substituting cy into the geodesic equation (4.6) and letting t = 0 yields

yi yk [)2 ( exp )i (0) fJyJfJyk

+ 2Gi(y) = 0.

In other words, Gi(y) are quadratic in y E TxM . Thus G is affine. We have proved the following

CHAPTER 14. EXPONENTIAL MAPS

224

Theorem 14.1.3 (Akbar-Zadeh (AZ2]) Let (M, G) be a spray space. exp is C 2 at zero sections in T M if and only if G is affine.

Recall that a spray G is said to be positively complete if every geodesic defined on (0, a) can be extended to a geodesic defined on (0, oo). By the above arguments, we see that if G is positively complete, then exp is defined on the whole T M. Further, exp is coo on T M \ { 0} and C 1 on the zero sections of TM by Theorem 14.1.1. In particular, expx: TxM-+ M is coo on TxM \ {0} and C 1 at the origin 0 E TxM. The question is whether or not expx covers the whole manifold M? Theorem 14.1.4 (Hopf-Rinow) Let (M, F) be a positive definite Finster space. The following are equivalent:

(a) F is positively complete; (b) expx : TxM-+ M is an onto map for some x E M; (c) expx : TxM-+ M is an onto map for any x EM; (d) every two points x 1 , x 2 E M can be joined by a minimizing geodesic.

The proof is omitted. See (BaChSh1]. The Hopf-Rinow Theorem is, however, not true for positively complete spray spaces. The following example shows that there exist positively complete sprays for which the exponential map expx TxM-+ M is not onto. Example 14.1.1 Consider the following spray on R 2 G =

u~ + v~- v)u 2 + v 2 ~ + u)u 2 + v2~. ax

ay

au

By Example 4.1.3, the geodesics of G are circles of radius r

av

(14.5)

= 1 on R 2 .

cos(.\t +e)- cose + x(O), sin(-\t +e)- sine+ y(O).

x(t) y(t)

Thus G is a complete spray. Express exp 0 : T0 R 2 = R 2 -+ R 2 by exp 0 (u,v)

= ((u,v) , 'll(u,v)) .

We obtain (u, v)

'll(u, v)

.Ju2 .Ju2

1

(v cos )u 2 + v 2 + usin )u 2 + v 2 - v),

1

(-ucos)u2+v2+vsin)u 2 +v 2 +u). (14.7)

+ v2

+ v2

(14.6)

Note that geodesics issuing from the origin form a disk of radius p = 2. This disk is the image of T0 R 2 = R 2 under exp 0 . In other words, exp 0 does not cover R 2 , although G is defined on the whole R 2 .

225

14.2. JACOBI FIELDS IN SPRAY SPACES

Suppose that the spray G in (14.5) is globally pointwise projective to a positive definite Finsler metric F on R2 . Since the geodesics of G and F are circles, F is complete. By the Hopf-Rinow Theorem, the exponential map ofF (which is also the exponential map of G) must cover the whole manifold. This is impossible. Thus G can not be globally pointwise projective to any positive definite Finsler metric on R 2 . Since G is isotropic, according to [GrMu], it is locally Finslerian, i.e., it can be locally induced by a (possibly not positive U definite) Finsler metric.

14.2

Jacobi Fields in Spray Spaces

In this section, we will discuss some basic properties of Jacobi fields induced by the exponential map. Let G be a positively complete spray on ann-manifold M. At every point xEM, expx: TxM-+ M is defined on the whole TxM. Take an arbitrary vector y E TxM \ {0}. The curve c(t) := expx(ty) is a geodesic issuing from x. For a vector v E TxM, consider the following special geodesic variation of c

H(s,t)

:=

expx (t(y

+ sv)) .

By Lemma 8.1.1, we know that (14.8)

CHAPTER 14. EXPONENTIAL MAPS

226

is a Jacobi field along c(t), 0

< t < oo. Namely, it satisfies DcDcJ + R c(J) = 0.

Since expx is C 0 at t = 0.

coo

(14.9)

on TxM \ {0} and C 1 at the origin, J is

coo on

(0, oo) and

t(y

Lemma 14.2.1 Let c: (-6,oo)-----+ M be a geodesic with c(O) = y E TxM. For any v E TxM, where 6 > 0, the Jacobi field J(t) := d(expx)ity(tv) , 0 < t::; oo, can be extended smoothly across t = 0 to a Jacobi field J on (- 6, oo) such that J(O) = 0,

(14.10)

DyJ(O) = v.

Proof By assumption, J satisfies (14.9) . Fix a number t 0 E (0, oo) . Since R c is

coo on [-6, oo), there exists an unique solution J of (14.9) on [-6, oo) satisfying

the initial data:

By the Uniqueness Theorem in O.D.E. theory, we have

](t)

= J(t),

t E (O,oo) .

In other words, J can be extended to a Jacobi field Since J(O) = 0, we can write

](t)

= tW (t),

J smoothly across t =

-6 ::; t ::; oo,

where W(t) is a C 00 vector field along c. Observe that for 0

0.

(14. 11)

< t < oo,

tW(t) = d(expx)ity(tv). Thus

W(t)

= d(expx)ity(v),

0

< t < oo.

(14.12)

14.2. JACOBI FIELDS IN SPRAY SPACES

227

By Theorem 14.1.1, we know that d(expx)lo =identity. Thus

W(O) = lim d(expx)ity(v) = v t-tO+

Differentiating (14.11) yields D.;J(t)

= W(t) + tD.;W(t).

Thus

Dyl(O) = W(O) = v. Q.E.D. By Lemma 14.2.1, we obtain the following

Lemma 14.2.2 Let c(t) := expx(ty), 0 ~ t < oo. For a number r > 0, expx is singular at ry if and only if there exists a Jacobi field J t- 0 along c, 0 ~ t ~ r, such that (14.13) J(O) = 0 = J(r).

Definition 14.2.3 Let (M, G) be a positively complete spray space. A vector y E Tx M \ {0} is called a conjugate vector of x if d(expx)ly : Ty(TxM) -+ Texp,(y)M is singular, and d(expx)lty is regular for 0 < t < 1. Denote by Conj(x) the set of all conjugate vectors at x. When G is induced by a Finsler metric L, for a vector y E TxM with L(y) = ±1, denote by cy > 0 the number such that CyY is a conjugate vector. We call cy the conjugate value of y. Given a spray space (M, G) , we may ask the following question: Under what condition on the geometric quantities of G, expx is non-singular on TxM? Let c(t) = expx(ty), 0 ~ t < oo, be a geodesic issuing from x and J(t) be a Jacobi field along c. Take a parallel frame {E;(t)}b 1 along c, namely, D.;E;(t) = 0 and {E;(t)}j= 1 is a basis for Tc(t)M for all t. Write

and Equation (14.9) becomes d2 Ji

dt 2 (t)

.

+ Rk(t)J

k

(t)

= 0.

(14.14)

If all solutions (Ji(t)) to (14.14) with (Ji(O)) = 0 and (ji(O)) =f 0 do not vanish at any t > 0, then expx is non-singular along the ray ty, t > 0.

CHAPTER 14. EXPONENTIAL MAPS

228

14.3

Comparison Theorems for Finsler Spaces

In this section, we will discuss geometric meaning of the sign of the Riemann curvature. Only positive definite Finsler metrics will be considered. We denote a positive definite Finsler metric L on a manifold M by

F(y)

:=

V£{0,

yETM.

F has non-positive curvature K ::; 0 if for any flag (P, y) C TxM, the flag curvature is nonpositive K(P, y) ::; 0.

Using Proposition 8.2.1, we obtain the following important results for Jacobi fields along a geodesic. Lemma 14.3.1 (Gauss Lemma) Let (M, F) be a Finster space and c: (a, b) -+ M a geodesic. Let J be a Jacobi field along c. Let y = c(t 0 ) , u = J(to) and v = DyJ(t 0 ), where t 0 E (a, b) . Then

9c(t)(J(t),c(t)) = gy(v,y)(t - to) + gy(u,y) .

(14.15)

Proof Since cis geodesic, Dec= 0. By (7.11), (8.2) and (8.57), we have :t (gc(J,

c))

~22 (gc(J,c))

gc(DtDcJ, c) -gt(Rt(J) , c) -gt(Rt(c), J) = 0.

Q.E.D.

This gives (14.15).

The orem 14.3.2 (Cartan-Hadamard [A us]) Let (M, F) be a positively complete Finster space. Suppose that F has nonpositive curvature K ::; 0. Then expx is non-singular on TxM for any x EM.

Proof Let y E TxM be an arbitrary unit vector and c(t) = expx(ty), 0::; t < oo. For a vector v E TxM, let J(t) := d(expx)jty(tv). J is the Jacobi field along c with J(O) = 0 and Dt J(O) = v . By the Gauss Lemma, we have J(t) = J J.(t) +(at+ b)c(t), where J l. is a Jacobi field gc-orthogonal to c. Note t hat J = JJ. if and only if v is gy-orthogonal toy. In order to study the zeros of J , it suffices to study that of J.l.. Thus we may assume that v is gy-orthogonal to y . Let

f

:=

Jgc(J, J).

14.3. COMPARISON THEOREMS FOR FINSLER SPACES Let r 0 > 0 and assume that J(r) satisfies

!

> 0 for 0 < r < r I=

g;;(DcJ, J)

f

0 •

229

Observe that on (0, r 0 ), f

.

The Schwarz inequality gives

(!') 2 ~ g;; (D ;; J, D;;J). We also have

(14.16)

= tJgy(v, v) + o(t),

f(t) and

(14.17) It follows from (14.16) and (14.17) that

gc(r)(D;;J(r), J(r))

f(r)j'(r)

for { g;;(DcJ, DcJ)- g;; (R;;(J) , J) }dt

> This implies

r(!')2dt?. f(r)2. r

lo

li(ln f(r))?. 0.

r dr Hence the quotient f(r)/r is non-decreasing on (0, r 0 ). We obtain 0

< r < ro.

From (14.19), we conclude that J(r) = d(expx)iry(rv) -:/:- 0 for all 0 Hence expx is non-singular along ry, 0 < r < oo.

(14.18)

(14.19)

< r < oo. Q.E.D.

The Cartan-Hadamard theorem indicates that for a positively complete Finsler space with nonpositive curvature, expx : TxM -+ M is a local diffeomorphism. With further arguments, we see that expx is actually a projection map or covering map . Thus if M is simply connected, then expx is a diffeomorphism. In this case, every pair of points p, q E M can be joined by an unique minimizing geodesic. See [BaChSh1] for more details. Our next goal is to study positively complete Finsler spaces with uniformly positive (Ricci) curvature. First, we introduce the index form for vector fields along a geodesic. Let c(t), 0 ~ t ~ r be a unit speed geodesic in a Finsler space (M, F) . For a vector field V(t) along c, define

CHAPTER 14. EXPONENTIAL MAPS

230

Observe that if E(t) is a parallel vector field along c with gc(E, E) = 1 and f(t) is a c= function with f(O) = 0, then

Lemma 14.3.3 Let y E TxM be a unit vector and c(t) := expx(ty). Suppose that expx is non-singular at ty E Tx M for all 0 < t ~ r. Then for any Jacobi field J(t) and any vector field V(t) along c with

V(O) = J(O) = 0, the index forms satisfy

V(r) = J(r),

I(V, V) ?:. I(J, J).

(14.20)

I(V, V) > 0.

(14.21)

The equality holds if and only if V = J. Hence for any non-trivial vector field V(t) along c with V(O) = V(r) = 0,

The proof is elementary, so it is omitted. See [BaChSh1]. Now we assume that expx is non-singular at ty for all 0 < t ~ r. Take an arbitrary basis {ei}i=l for TxM with e1 = y and {J;(t)}~ 1 the Jacobi fields along c with J;(O) = 0 and Dcl;(O) = e;. Fix an arbitrary function f(t) with f(O) = 0 = f(r). By Lemma 14.3.3, for each i = 2, · · · , n, (14.22) Adding them together, one obtains

1r {

(n -1)(!') 2

-

Ric(c)f 2 }dt > 0.

(14.23)

Assume that Ric(y)?:. (n- 1)F2 (y). Then (14.23) is simplified to (14.24) (14.24) holds for any c= function f =f. 0 with f(O) = 0 = f(r). We claim that 0 < r ~ 1r. Suppose this claim is false. Then for some c: > 0, expx is non-singular at ty for all 0 < t ~ r := (1 + c:)1r. Let

f(t) =sin

(-t-). 1+c:

Plugging f(t) into (14.24), one obtains (1 + c:) [ (1 : c:) 2

-

1]

=for {(1 :

c:) 2 cos 2

C: J - C: sin 2

c;) }dt > 0.

Clearly, this is impossible, because that the left hand side is negative. Thus the claim is true. We have proven the following

14.4. JACOBI FIELDS IN ISOTROPIC SPRAY SPACES

231

Theorem 14.3.4 (Bonnet-Myers [A us]) Let (M, F ) be ann-dimensional posi-

tively complete Finsler space. Suppose that Ric(y) 2:: (n- 1)F2 (y),

\/y E TM \

{0}.

Then for any unit vector y E Tx M, the conjugate value satisfies Cy :S

1r .

Let (M, F) be a Finsler space as in Theorem 14.3.4. For a point x E M, define Diam(M, x) := sup d(x, z) . zEM

We can show that Diam(M,x) :S

1r.

Consider the lifted Finsler metric on the universal covering ( M, x) -t ( M, x). F is also positively complete and satisfies the same Ricci curvature bound. Thus Diam(M, x) :S

1r.

In particular, M is compact. We conclude that the fundamental group of M is finite. See [BaChSh1] for more details.

14.4

Jacobi Fields in Isotropic Spray Spaces

One of the important problems in Spray Geometry is to understand the relationship between the Riemann curvature and the global geometrical/topological structures of a spray space. A natural approach is to study the behavior of the exponential map. When the spray is isotropic, the relationship between the Riemann curvature and the exponential map becomes much simpler. Let (M, G) be a n isotropic spray space. Namely, the Riemann curvature Ry has the following form

Ry(u) = R(y)u + Ty(u) y, where Ty E

r; M

satisfies

Ty(y)

(14.25)

= - R(y).

(14.26)

Let c(t) be a geodesic of G and R (t) := R(c(t)) . Let s(t) and c(t) denote the solutions of the following equation

y"(t) with s(O)

= 0,

s' (0)

+ R(t)y(t)

= 1,

= 0,

c(O) = 1, c' (0) = 0.

(14.27)

CHAPTER 14. EXPONENTIAL MAPS

232

Lemma 14.4.1 For any Jacobi field J along c, there are two parallel vector fields E(t) and E*(t) along c, which are linearly independent of c(t), such that

where

0.

This proves the following Theorem 14.4.2 Let (M, G) be a positively complete isotropic spray space. Suppose that Ric :S 0.

Then for any x EM , expx :TxM-+ M is a local diffeomorphism.

Now we consider the case when Ric > 0. In general, from (14.27), we do not know whether or not the solution s(t) has a zero in (0, oo). Suppose that s(t) = 0 for some t > 0. Let r denote the first zero of s(t) . It follows from (14.33) that

d(expx)Jry(v) =-[for

1 8

Tc(p)(E(p))s(p) dpds] c(r) ,

where E(t) is the parallel vector field along c with E(O)

= v.

Let

e(v) :=-for ~a· Tc(p) ( E(p) )s(p) dpds . Then 6 : TxM-+ R is a linear form. It follows from (14.26) that

Tc(c)

= -R(c).

(14.34)

CHAPTER 14. EXPONENTIAL MAPS

234 Thus

G(y)

=for los R(c(p))s(p) dpds > 0.

Let

Wy := ker0. The above argument shows that Wy is a hyperplane andy fl. Wy· From (14.34), we see that d(expx)Jry = 0 on Wy. This proves (i) and (ii) in the following Theorem 14.4.3 Let (M, G) be positively complete isotropic spray space. Suppose that Ric> 0.

Then for a non-zero vector y E Tx M, one of the following holds (i) expx is non-singular along x(t) = ty, 0 :S t < oo, (ii) there is a number r > 0 and a hyperplane W y C Tx M such that d( expx) Jry = 0 on Wy. If along c(t) = expx(ty), t 2: 0, Ric(c(t)) 2: (n- 1)>. > 0.

Then only case (ii) occurs at r :S

(14.35)

:fi_.

Proof: Suppose the Ricci curvature satisfies the bound (14.35) along a geodesic c. Then the function R(t) := R(c(t)) in (14.27) satisfies

>. > 0.

R(t) 2:

Assume that s(t) is positive on an interval (0, r). By an elementary argument, one can show that the solution s(t) satisfies ()

s t

This implies r

:S 1r / ..fi.

< -

sin(..fit) 1\ v>.

'

\10

< t:::;

r.

Q.E.D.

Recall that a spray G is said to be weakly Ricci-constant if the Ricci curvature Ric= (n- 1)R satisfies (14.36) By Lemma 8.1.9, the Ricci curvature Ric(c(t)) = >. is constant along any geodesic c. By the same argument as for Theorem 14.4.3, we immediately obtain the following

14.4. JACOBI FIELDS IN ISOTROPIC SPRAY SPACES

235

Corollary 14.4.4 Let (M, G) be a positively complete isotropic spray space. Suppose that G is weakly Ricci-constant. Then for any y E TxM \ {0} with R(y) = 1, there is a hyperplane Wy C TxM such that

(14.37)

Example 14.4.1 Consider the following spray on R 2

2 + v2 ~ 2 2 G=u~+v~ av au ax ay -vJu +v ~+uVu

(14.38)

A direct computation using (8.42) or (8.44) yields

R1

_ 1-

v2

R2 -_ -vu, 1

R 21 = -uv,

'

Thus the Ricci curvature is given by R(u,v) = R~ +R~ = u 2 +v 2

> 0.

(14.39)

Moreover, G is weakly Ricci-constant, i.e.,

+ v2

R ;mYm = Rxu + Ryv- vVu 2

Ru

+ uVu2 + v 2

R v = 0.

By Corollary 14.4.4, we know that exp0 is singular on the tangent sphere u 2 + v 2 = 1r2 . We are going to verify this fact directly. By Example 14.1.1, the exponential map exp 0 : T0 R 2 = R 2 -+ R 2 is given by exp0 (u, v) = ( (u, v), 'lt(u , v) ) , where (u, v) 'lt(u, v)

../u2

+ v 2 + usin Ju 2 + v 2 -

1

( v cos Ju2

1

(- ucos Ju 2 + v 2

+ v2

vu2 + v2

v) ,

(14.40)

+ v sin Vu 2 + v 2 + u) .(14.41)

A direct computation yields \lf u

v

_ \lf _ v

u-

. v'U 2 + V 2 ../u2 + v2

Sill

Thus exp0 is singular along circles of radius k1r in T0 R 2 = R 2 , where k = 1, 2, · · ·. Observe that 2 exp0 (u,v) = -(- v,u), 7f

exp0 maps the circle of radius r p = 2 in R 2 •

= 7f

in T0 R 2

= R2

onto the circle of radius

CHAPTER 14. EXPONENTIAL MAPS

236

...

..

According to Lemma 8.1.10, two-dimensional sprays are always isotropic. Thus we can apply the above results to two-dimensional sprays associated with a SODE. Let q, = q, (x, y, ~) be a coo function on R 2 x R. Consider the following equation on R dy) ( d2 y (14.42) ds 2 = q, x,y, dx · To study (14.42), we construct a homogeneous system on R 2 {

d2x (it'l"

dx if:JL) + 2G( X' y' dt ' dt -

0

d2y

~J:JL) dt ' dt + 2H ( X' y' dx

0

dt'i

(14.43)

where G and H satisfy q,(x , y,~)

= 2~G(x,y, 1,~)- 2H(x,y, 1,~).

(14.44)

By Lemma 3.1.1, we know that the graphs of solutions of (14.42) in R 2 coincide with the solutions of (14.43) in R 2 as point sets. Thus, we can study the solutions of (14.42) by investigating the geodesics of the corresponding spray on R 2 8 G := u OX

8

+ v By

8 8 - 2G(x, y, u, v) au - 2H(x, y, u, v) 8v,

(14.45)

By (8.42), the Ricci curvature R =Ric is given by R

=

+ 2Hy- a;,- H;- 2HuGv- u(Gu + Hv)x -v(Gu + Hv)y + 2G(Gu + Hv)u + 2H(Gu + Hv)v·

2Gx

(14.46)

Suppose that G is a positively complete spray on R 2 with (3.13) such that the Ricci curvature Ric= R::; 0. By Theorem 14.4.2, we know that for any x E R 2 , the exponential map exp(x,y) : T(x ,y)R 2 -+ R 2 is a local diffeomorphism.

14.4. JACOBI FIELDS IN ISOTROPIC SPRAY SPACES

237

Example 14.4.2 Consider a equation (14.42) where ~(x, y, ~) := A(x, y)

+ B(x, y)~ + C(x, y)e + D(x, y)e,

(14.47)

where A,B,C,D are coo functions on R 2 . There are several homogeneous systems (14.43) which we can use to study the equation (14.47) by Lemma 3.1.1. Below are some natural choices for (14.43) . Option 1: We take = ~D(x, y)v 2

{~

-~ ( A(x, y)u 2 + B(x, y)uv + C(x, y)v 2 )

(14.48)

By (14.46), we obtain

2 ( AC + ~B 4 2x - ~B

R =

- A y)u2 + (2AD- ~B 2y + C x )uv (14.49)

+GBD +Dx ) v 2 . Option 2: We take

{

H

B(x, y)u 2

+ C(x, y)uv + D(x, y)v 2 )

G:

=

H:

= - ~A(x,y)u 2 .

(14.50)

By (14.46), we obtain R

GAC-Ay)u 2 +GCx +2AD-By)uv

=

2 )v 2 . +(D - ~c 4 2 y + BD- ~c

(14.51)

X

Option 3: We take

{G H

= ~B(x, y)u2 + ~C(x, y)uv + ~ D(x, y)v 2 = -~A(x,y)u 2 - ~B(x,y)uv- ~C(x,y)v 2

(14.52)

By (14.46), we obtain

R

=

2 2 1 ( -B -AY - -B 9 3 x

) u 2 + (2-C 2 + -AC 3x 3

22 1 4 + 2AD ) uv + ( D X - -C --BC 9 3 y - -C 9

2 -B 3Y

)2v . 2 + -BD 3

(14.53)

Option 4: Assume that D = 0. Besides the above options, we can also take

{ G: H:

= ~B(x,y)u 2 + C(x,y)uv = -~A(x, y)u 2 + ~C(x, y)v 2 •

(14.54)

238

CHAPTER 14. EXPONENTIAL MAPS

By (14.46), we obtain (14.55) For any option, if R :S 0, then by Theorem 14.4.2, we know that for any x E R 2 , the exponential map of the corresponding spray G is a local diffeomorphism.

exp(x ,y) : T(x,y)R 2 -+ R2 .

14.5

Finsler Spaces of Constant Curvature

Although there are infinitely many positively complete Finsler spaces of constant curvature A, the topology of the manifold M is quite special. By Theorem 14.4.2 we know that if A :S 0, the exponential map expx : TxM-+ M is a local diffeomorphism. By Proposition 14.4.3, we know that if A > 0, the exponential map expx : TxM -+ M is singular in all directions, provided that the Finsler metric L > 0 on T M\ { 0}. In this section, we will apply the techniques developed in the previous sections to study Finsler spaces of positive constant curvature. Proposition 14.5.1 Let (M, L) be a positively complete Finster space of constant curvature A::/; 0. For a point x E M, let

~x := {y

E TxM, AL(y) = 1 }·

Then for any connected component

~~ C ~x,

expx ( 7!"~~) = x*.

M

(14.56)

14.5. FINSLER SPACES OF CONSTANT CURVATURE Proof For a vector y E

~x,

239

let

yl. :=

{wE TxM,

gy(y, w) = 0 } ·

Since

L(y) = 9y(y, y) "/; 0, yl. is a hyperplane which does not contain y . For an arbitrary vector w E TxM satisfying d(expx)l,.y(w) = 0, put w = w1

+ ry,

Let J1(t) := d(expx)lty(twl) It follows from the Gauss Lemma that

Thus 9c(7r) ( d(expx)l,.y(w), c(7r))

0

9c(7r) (Jl(7r)

+ rc(7r), c(?r))

>.L( c( 1r)). This implies r = 0. Thus w E yl.. That is,

Take an arbitrary curve y(s) in

~x

and let

Then This implies

&(s) = d(expx)l,.y(s) (y'(s)) = 0. Thus

O"(s) =a point. Q.E.D.

Let (M,L) be as in Proposition 14.5.1. Let

~;

:= { y E TxM,

>.L(y) = -1 }·

(14.57)

CHAPTER 14. EXPONENTIAL MAPS

240

By a similar argument, one can prove that expx (t~;) grows exponentially. More precisely, let v E Ty~;, then d(expx)ity(tv)

= sinh(t) E(t),

(14.58)

where E(t) is a parallel vector field along c(t) := expx(ty) with E(O) = v.

Example 14.5.1 Consider a Finsler metric defined on M 2 = ( -oo, oo) x S1 , L(x , y , u,v) = - -1-2 u

1+x

2+ (1 + x 2) v 2.

(14.59)

The Gauss curvature K=l.

Since Lis a pseudo-Finsler metric, (M, L) looks like a surface in the Euclidean space lR 3 with negative curvature.

If a Finsler metric is positive definite, then the positivity of the Riemann curvature makes the space compact. Proposition 14.5.2 Let (M, F) be a positively complete positive definite Finsler space of constant curvature R = 1. Then for any x E M, there is a point x* E M such that (14.60) expx SxM) = x*,

(1r

where SxM denotes the unit sphere of Fx in TxM.

14.5. FINSLER SPACES OF CONSTANT CURVATURE

241

Proof Since F is positive definite, the unit sphere

is a strongly convex closed hypersurface around the origin. Thus the set :Ex in (14.56) is the whole indicatrix

It follows from Proposition 14.5.1 that there is a point x* EM such that expx ( 1rSxM) = x*. This implies that M must be compact with d(x, x*) :::; 1r.

Q.E.D.

x*

Let F be a positive definite Finsler metric on sn with constant curvature R = 1. Fix a point x E gn. There is an unique point x* E gn with d(x, x*) = 1r. The exponential map expx : TxSn -+ sn gives rise to a diffeomorphism

cp: (0,1r) by

X

SxM-+ gn- {x,x*}

cp(t,y) := expx(ty).

The induced Riemannian metric 9x on SxM is given by

9x(u,v)

:=

gy(u,v),

Let

Y:=cp.(%t).

CHAPTER 14. EXPONENTIAL MAPS

242

y is a unit geodesic field on

sn from X to x*.

Define

g := gy. g is a smooth Riemannian metric on

sn - {X' x*}.

We claim that (14.61)

Proof Fix y E SxM and let c(t)

:=