One of the most cited books in mathematics, John Milnor's exposition of Morse theory has been the most important bo
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Table of contents :
CONTENTS
PREFACE
PART I. NONDEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD
§1. Introduction
§2. Definitions and Lemmas
§3. Homotopy Type in Terms of Critical Values
§4. Examples
§5. The Morse Inequalities
§6. Manifolds in Euclidean Space: The Existence of Nondegenerate Functions
§7. The Lefschetz Theorem on Hyperplane Sections
PART II. A RAPID COURSE IN RIEMANNIAN GEOMETRY
§8. Covariant Differentiation
§9. The Curvature Tensor
§10. Geodesics and Completeness
PART III. THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS
§11. The Path Space of a Smooth Manifold
§12. The Energy of a Path
§13. The Hessian of the Energy Function at a Critical Path
§14. Jacobi Fields: The Nullspace of E**
§15. The Index Theorem
§16. A Finite Dimensional Approximation to Ω^c
§17. The Topology of the Full Path Space
§18. Existence of Nonconjugate Points
§19. Some Relations Between Topology and Curvature
PART IV. APPLICATIONS TO LIE GROUPS AND SYMMETRIC SPACES
§20. Symmetric Spaces
§21. Lie Groups as Symmetric Spaces
§22. Whole Manifolds of Minimal Geodesics
§23. The Bott Periodicity Theorem for the Unitary Group
§24. The Periodicity Theorem for the Orthogonal Group
APPENDIX. THE HOMOTOPY TYPE OF A MONOTONE UNION
Annals of Mathematics Studies Number 5 1
MORSE T H E O R Y BY
J. Milnor Based on lecture notes by M. SPIVAK and R. WELLS
PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS
Copyright © 1963, © 1969, by Princeton University Press All Rights Reserved L.C. Card 6313729 ISBN 0691080089 Third Printing, with corrections and a new Preface, 1969 Fourth Printing, 1970 Fifth Printing, 1973
Printed in the United States of America 19
18
17
16
15
14
PREFACE This book gives a presentday account of Marston Morse's theory of the calculus of variations in the large.
However, there have been im
portant developments during the past few years which are not mentioned Let me describe three of these R. Palais and S. 3male have studied Morse theory for a realvalued function on an infinite dimensional manifold and have given direct proofs of the main theorems, without making any use of finite dimensional ap proximations.
The manifolds in question must be locally diffeomorphic
to Hilbert space, and the function must satisfy a weak compactness con dition. M
As an example, to study paths on a finite dimensional manifold
one considers the Hilbert manifold consisting of all absolutely con
tinuous paths
or. [o,l 3 > M with square integrable first derivative. Ac
counts of this work are contained in R. Palais, Morse Theory on Hilbert Manifolds, Topology, Vol. 2 (1963), pp. 29931*0; and in S. Smale, Morse Theory and a Nonlinear Generalization of the Dirichlet Problem, Annals of Mathematics, Vol. 80 (196I+), pp. 382396. The Bott periodicity theorems were originally inspired by Morse theory
(see part IV). However, more elementary proofs, which do not in
volve Morse theory at all, have recently been given.
See M. Atiyah and
R. Bott, On the Periodicity Theorem for Complex Vector Bundles,Acta Mathematica, Vol. 112 (1964), pp. 2292t7, as well as R. Wood, Banach Algebras and Bott Periodicity, Topology, b (196566), pp. 3 7 1 3 8 9 Morse theory has provided the inspiration for exciting developments in differential topology by S. Smale, A. Wallace, and others, including a proof of the generalized Poincare hypothesis in high dimensions.
I
have tried to describe some of this work in Lectures on the hcobordism theorem, notes by L. Siebenmann and J. Sondow, Princeton University Press, 1965.
Let me take this opportunity to clarify one term which may cause fusion.
In §12 I use the word "energy" for the integral v
con
vi
PREFACE
E=y
0
along a path a>(t) . V. Arnol'd points out to me that mathematicians for the past 200 years have called E the "action”integral. This discrepancy in terminology is caused by the fact that the integral can be interpreted, in terms of a physical model, in more than one way. Think of a particle P which moves along a surface M during the time interval 0< t < 1 . The action of the particle during is defined
this time interval
to be a certain constant times the integral E.
If no forces
act on P (except for the constraining forces which hold it within M), then the "principle of least action" asserts that E will be minimized within the class of all paths joining a>(o) to 03(1), or at least that the first variation of E will be zero.
Hence P must traverse a geodesic.
But a quite different physical model is possible.
Think of a rubber
band which is stretched between two points of a slippery curved surface. If the band is described parametrically by the equation x = o)(t), 0 < t < 1 , then the potential energy arising from tension will be proportional to our integral E (at least to a first order of approximation). For an equilibrium position this energy must be minimized, and hence the rubber band will describe a geodesic. The text which follows is identical with that of the first printing except for a few corrections.
I am grateful to V. Arnol'd, D. Epstein
and W. B. Houston, Jr. for pointing out corrections. J.W.M.
Los Angeles, June 1968.
CONTENTS PREFACE
v
PART I.
NONDEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD § 1. Introduction.........................................
1
§2. Definitions and Lemmas...............; ..............
k
§3 . Homotopy Type in Terms of Critical Values.............
12
§^. Examples............................................
25
§5 . The Morse Inequalities...............................
28
§6. Manifolds in Euclidean Space:
The Existence of
Nondegenerate Functions
32
§7 . The Lefschetz Theorem on Hyperplane Sections...........
PART II.
39
A RAPID COURSE IN RIEMANNIAN GEOMETRY §8. Covariant Differentiation............................
^3
§9. The Curvature Tensor.................................
51
§10. Geodesics and Completeness............................
55
PART III. THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS .§11. The Path Space of a Smooth Manifold...................
67
§12. The Energy of a Path.................................
70
§13. The Hessian of the Energy Function at a Critical Path . .
7^
§D.
77
Jacobi Fields:
The Nullspace of
E**................
§1 5 . The Index Theorem...................................
83
§1 6 . A Finite Dimensional Approximation to Qc .............
88
§1 7 . The Topology of the Full Path Space...................
93
§1 8 . Existence of Nonconjugate Points ....................
98
§19. Some Relations Between Topology and Curvature ..........
100
vii
CONTENTS PART IV.
APPLICATIONS TO LIE GROUPS AND SYMMETRIC SPACES §20.
Symmetric S p a c e s ........................
§21.
Lie Groups as Symmetric S p a c e s
§22.
Whole Manifolds of Minimal Geodesics.
§23.
The Bott Periodicity Theorem for theUnitary Group
§2k .
The Periodicity Theorem for the OrthogonalGroup.. . . .
APPENDIX.
........ 109
................... 112 ............. 118 ...
12k 133
THE H0 M0 T0 PY TYPE OF A MONOTONE U N I O N .................... 1U9
viii
PART I NONDEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD.
§1 . Introduction. In this section we will illustrate by a specific example the situ ation that we will investigate later for arbitrary manifolds. sider a torus
M,
tangent to the plane
V,
Let us con
as indicated in Diagram 1.
Diagram 1 . Let above the V f(x) < a.
f: M —►R
(R always denotes the real numbers) be the height
plane, and let
M8, be the set of all points
x e M
such that
Then the following things are true: (1)
If a < 0 = f (p), then
(2)
If f(p) < a < f(q),
then
M? is homeomorphic to
a 2cell.
(3 ) *
If f(q) < a < f(r), then
M8,is homeomorphic to
a cylinder:
(i*)
M8, is vacuous.
If f(r) < a < f(s), then M8, is homeomorphic to a compact manifold of genus one having a circle as boundary:
1
2
I. NONDEGENERATE FUNCTIONS
(5 )
If
f(s) < a,
then
M8
is the full torus.
In order to describe the change in of the points
f(p),f(q),f(r),f(s)
as
a
passes through one
it is convenient to consider homotopy
type rather than homeomorphism type. (1) —► (2)
M8
In terms of homotopy types:
is the operation of attaching a ocell.
homotopy type is concerned, the space
For as far as
M8, f(p) < a < f(q), cannot be dis
tinguished from a ocell:
■ Here
"«"
means "is of the same homotopy type as." (2) ► (3)
(3)
is the operation of attaching a
1cell:
(J+) is again the operation of attaching a 1cell:
(U) *• (5)
is the operation of attaching a 2_cell.
The precise definition of "attaching a kcell" can be given as follows.
Let
Y
be any topological space, and let ek
=
{x e R k : x < 1}
be the kcell consisting of all vectors in Euclidean kspace with length
 0 • The real number
f(p)
is called a critical value of
We denote by If
a
M8, the set of all points
is not a critical value of
function theorem that f~1(a)
f
f.
x € M
such that
then it follows from the implicit
M8, is a smooth manifoldwithboundary.
is a smooth submanifold of A critical point
p
f(x) < a.
The boundary
M.
is called nondegenerate if and only if the
matrix
.o a2f (p)) dx^dx^
is nonsingular.
It can be checked directly that nondegeneracy does not
depend on the coordinate system.
This will follow also from the following
intrinsic definition. If functional then
v
p
f**
and
w
is a critical point on
TMp,
where
f we define a symmetric bilinear
called the Hessian of
have extensions
f**(v,w) = vp(w(f)),
of
vp
v
and
w
f
p.
If
to vector fields.
is, of course, just
this is symmetric and welldefined.
at
v.
v,w € TMp We let
We must show that
It is symmetric because
vp (w(f))  ftpWf)) = [v,w]p (f) = 0 where
[v,w]
Here w(f)
is the Poisson bracket of
v
and
w,
denotes the directional derivative of
and where [v,w] (f) = 0 f
in the direction w.
§2. since
f has
p
as a critical point.
Therefore
f**
Vp(w(f)) = v(w(f)) Wp(v(f))
DEFINITIONS AND LEMMAS
is symmetric.
It is now clearly welldefined since
is independent of the extension
v
of
v,
while
is independent of w. If
(x"*,...,xn) is a local coordinate system and v = E a. —^rl , ^ d 1 w = E b. —  _ we can take w = E b. — s where b. now denotes a conJ p J J n
stant function.
Then
f**(v,w) = v(w(f))(p) = v(E b.
=
3 8xJ /
so the matrix
\
respect to the basis
—^r
a. b. — jMU (p) ; 1 J 8x 8x
E
ij
represents the bilinear function
f**
with
.. ., — _ p' ^ x n p
We can now talk about the index and the nullity of the bilinear functional tor space
f** on V,
on which
H
TM^.
The index of a bilinear functional
for every
is negative definite; the nullity is the dimension of the null
w e V. f
index of
f**
on a vec
is defined to be the maximal dimension of a subspace of V
space, i.e., the subspace consisting of all
point of
H,
The point
if and only if on
TMp
p f**
v e V
such that
H(v,w) = 0
is obviously a nondegenerate critical on
TM^
has nullity equal to
0.
will be referred to simply as the index of
The Lemma of Morse shows that the behaviour of described by this index.
f
at p
The f
can be completely
Before stating this lemma we first prove the
following: LEMMA 2.1 . Let f be a C°° function in a convex neigh borhood V of 0 in R n, with f(0) =0. Then n f(x1,...,xn) = ^ x ^ C x , , . . . ^ ) i= 1
for some suitable
C°° functions
defined in V,
with
si(0) = H i(o) * PROOF:
. P df(tx1,...,tx )
f (X1,•••,xn) =
J
0
at
n r V d t  J ^ ( t X i , . . . , t x n)Xidt. 0 i=i 1 1
Therefore we can let
gi(x1,... ,xn) =
J 0
( t x ^ ,... ,txn) dt .
1
at
p.
6
I. NONDEGENERATE FUNCTIONS LEMMA 2.2 (Lemma of Morse) . Let p be a nondegenerate critical point for f. Then there is a local coordinate system (y1,*,yn) in a neighborhood U of p with yi(p) = o for all i and such that the identity f . f(p) . (yV 
holds throughout
PROOF:
...  ( y V
U,
where
\
 (yV
is the index of
f
at
p.
We first show that if there is any such expression for
X must be the index of
then
♦ (yx+1)2 ♦
f
at
p.
f,
For any coordinate system
(z1,..., zn), if f(q) = f(p)  (z’(q))2 ...  (zx (q))2 + (zX+1 (q))2 + ... + (zn (q))2
then we have
—dzi^dzJ r(P) =
X ’ otherwise ,
which shows that the matrix representing I 3 7
IP’
± 
i =J < x , ,
f**
with respect to the basis
is
’ Sz11 lp
/
>
("'■.) Therefore there is a subspace of
TMp
tive definite, and a subspace V
of dimension
definite.
If there were a subspace of
on which
f** were
which is
clearlyimpossible.
x
of dimension
TM^
negativedefinite then
nx
where
where
p
is positive x
this subspace would intersect V, of f**.
We now show that a suitable coordinate system Obviously we can assume that
f**
is nega
of dimension greater than
xis the index
Therefore
f**
is the origin of R n
(y1,...,yn) exists. and that
f(p) = f(o) = o*
By 2.1 we can write n f(Xl ,...,xn ) = £
xjgj(x1,...,xn )
j= 1 for
(x1,...,xn) in some neighborhood of
0.
Since
critical point: sj(0) = ^ r r throughout
U 1; where the matrices
a linear change in the last Let
nr+i
(H^ (u1,... ,un)) are symmetric. After coordinates we may assume that
g(u1,...,un) denote the square root of
be a smooth, nonzero function of uj,...,^ borhood
U2 C Ui
vi = ui
of
0.
13^ ( 0 ) 4 0.
Hpr(u1,...,un). This will throughout some smaller neigh
Now introduce new coordinates
v1,...,vn
by
for 1 ^ r + Y ulH Lr(u1,...,un)/Hpi>(u1,...,un)l. 1> r
vr (u1,...,un) =
It follows from the inverse function theorem that
v1,...,vn will serve as
coordinate functions within some sufficiently small neighborhood It is easily verified that f ■ Y
± = = + 1.
Thus the corresponhence t — f( 0. Then, for all sufficiently small e, the set Mc+e has the homotopy type of Mc”e with a xcell attached.
The idea of the proof of this theorem is indicated in Diagram 4 , for the special case of the height function on a torus.
The region
Mce = fi is heavily shaded.
We will introduce a new function
coincides with the height function borhood of
p.
Thus the region
gether with a region
H
near
p.
f
F: M *■ R
except that F < f
F”1(oo,cs]
Diagram 4.
in a small neigh
will consist of Mc_e
In Diagram 4 , H
shaded region.
which
to
is the horizontally
§3. Choosing a suitable cell
HOMOTOPY TYPE e
C H,
15
a direct argument
ing in along the horizontal lines) will show that retract of Mc"e u H.
(i.e., push
Mc”eu e*' is a deformation
Finally, by applying 3.1 to the function F
region F”1[ce,c+s] we will see that
Mc_e u H
and the
is a deformation retract
of Mc+e. This will complete the proof. Choose hoose a c coordinate o< system u1,...,un
in a neighborhood
U
of
p
so that the identity f = c  (u1)2 ...  (ux)2 + (ux+1)2+... + (Un)2 holds throughout
U.
Thus the critical point
p
will have coordinates
u1(p)  ... « un (p) = 0 . Choose
e > 0 (1)
sufficiently small so that The region
points other than (2)
f~1[ce,c+e]
is compact and contains no critical
p.
The image of
U
under the diffeomorphic
(u\...,un):
U
imbedding
►R
contains the closed ball. ((u1,....u11) : Z ( u 1)2 s ii(r) = 0 1 < where
n ’(r) =
neighborhood
U,
for
r > 2e
ii*(r) < 0
. Now let
for all r,
F coincide with
f
outside of the coordinate
and let
F = f  n((u1)2+... +(uX)2 + 2(uX+1)2+...+2(un)2) within this coordinate neighborhood.
It is easily verified that
well defined smooth function throughout
F
is a
M.
It is convenient to define two functions 1 ,1: TJ
► t0,»)
by I = ( u V + ... + (ux)2 t, = (uX+1)2 + ... + (u11)2 Then
f = c  £ + t^;
so that:
F(q) = o  i(q) + n(q)  ix((q) + Sn(q))
for all
q e U. ASSERTION 1. Mo+e = fi(_ PROOF:
The region
F”1(
Outside of the ellipsoid
c+e] coincides with the region
 + 2t^ < 2e the functions
f and
ade
§3. F
coincide.
HOMOTOPY TYPE
17
Within this ellipsoid we have F < f = ct+n < c+ l+n < c+e .
This completes the proof. ASSERTION 2. PROOF:
The critical points of F
are the same as those of
f.
Note that  = 1  n'(! +2t)) < 0 9P ■Sri =
1  2u'(t +2t) > 1 .
Since
^ =H where the covectors it follows that
F
d
and
+ ^ d"
dti are simultaneously zero only at the origin,
has no critical points in
U
other than the origin.
Now consider the region F“1[ce,c+e].
By Assertion 1 together
with the inequality F < f we see that F”1[ce,c+e] C f“1[ce,c+e] . Therefore
this region is compact.
except possibly
p.
It can contain no criticalpointsof F
But F(p) = c  n(0) < c  e .
Hence
F“1[ce,c+e]
contains no critical points.
Together with 3.1 this
proves the following. ASSERTION 3 . The region
F~1(oo,ce]
is a deformation retract of
Mc+e. It will be convenient to denote this region Mc~e u H;
where
REMARK: described
as
H
F"1(0. CASE 2.
rQ maps the entire region into
F~1(°o,cs3
maps
Within the region
e\
The
into itself, follows from the in
6 < 
* < V ' % Z > which completes the proof.
■
I
(W
^
+
TT )
■ m < V ’W > 
II. RIEMANNIAN GEOMETRY COROLLARY Q.k. vector Xp €
For any vector fields
Xp
and completes the proof that
P
P
is parallel along
Henceforth we will assume that
M
is identically zero; s.]
is a Riemannian manifold, pro
vided with the unique symmetric connection which is compatible with its metric.
In conclusion we will prove that the tensor
R
satisfies four
symmetry relations.
LEMMA 9.3. The curvature tensor of a Riemannian manifold satisfies: (1 ) R(X,Y)Z + R(Y,X)Z = 0 (2 ) R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0 (3) + = 0 (1*) = .
The skewsymmetry relation (1 ) follows immediately from the
PROOF:
definition of R. Since all three terms of (2) prove (2) when the bracket products zero.
are tensors, it is sufficient to [X,Y], [X,Z]
and
[Y,Z]
are all
Under this hypothesis we must verify the identity  X b (Y b Z) +
Y b (X b Z)
Yb(ZbX)
Zb(YbX)
+
 Z b (X b Y) +
X b (Z b Y)
=
0 .
But the symmetry of the connection implies that Y b Z  Z b Y
=
= 0
[Y,Z]
Thus the upper left term cancels thelower right maining terms cancel in pairs.
.
term.Similarly the re
This proves (2).
To prove (3 ) we must show that the expression skewsymmetric in
Z and W.
This is clearly equivalent to the assertion
that
for all
X,Y,Z.
is
Again we may assumethat
=
0
[X,Yl
= 0,
so that
is equal to <  X h (Y b Z) + Y I (X H Z),Z>
•
54
II.
RIEMANNIAN GEOMETRY
In other words we must prove that the expression < Y b (X b Z),Z> is symmetric in X Since and
Y.
and
[X,Y]
Y.
=
0
the expression YX
is symmetric in X
Since the connection is compatible with the metric, we have X
2
hence YX < Z , Z > = 2 < Y H X I Z),Z> + 2 < X I Z,Y h Z > . But the right hand term is clearly symmetric in X
is symmetric in X
and
and
Y.
Therefore
Y; which proves property (3 ).
Property (4 ) may be proved from (1 ), (2), and (3 ) as follows.
►
r< R (Z , X)Y,
< R (X,W )Y,Z>
Formula (2) asserts that the sum of the quantities at the vertices of shaded triangle W
is zero.
Similarly (making use of (1 ) and (3 )) the
sum of the vertices of each of the other shaded triangles is zero.
Adding
these identities for the top two shaded triangles, and subtracting the identities for the bottom ones, this means that twice the top vertex minus twice the bottom vertex is zero.
This proves (4 ), and completes the proof
§1 0 . GEODESICS AND COMPLETENESS
§10. Let
M
Geodesics and Completeness
be a connected Riemannian manifold.
DEFINITION.
A parametrized path 7:
where
I
55
I  M,
denotes any interval of real numbers, is called a geodesic
acceleration vectorfield vector field identity
^ must
^
^ isidentically
beparallelalong
d . d,
dr .
y.
2 . Ddr
y.
Thus thevelocity
If y is ageodesic,
of the velocity vector is
Introducing the arclength function
s(t)
=
^
+ constant
This statement can be rephrased as follows:
The parameter t
geodesic is a linear function of the arclength. ally equal to the arclength if and only if
along
The parameter
^ =
t *• r(t) e M
The equation pjr ^
determines
n
smooth functions
is actu
u1,*,un
u 1(t),...,un(t).
for a geodesic then takes the form
■d2uk . y
rk
, 1
L
rij
(u ' •• •’U 3“TO ~ 5Z
+
dt
t
a
1
In terms of a local coordinate system with coordinates a curve
then the
dr .
shows that the length ^ = !  ' ( « I2 where equality
holdsonly if
^
b 1 a
= 0;
henceonly if
^ =
0.
Thus
b [^dt > y
r'(t) dt > r(b)  r(a) 1 a
where equality holds only if
r(t)
is monotone and
v(t) is constant.
This completes the proof. The proof of Theorem 10.1* piecewise smooth path
©
from
q
qf v
expq (rv) e Uq ;
=
o < r < s,
tain a
segment joining the spherical shell of radius r,
segment will be
> r  5;
will be
> r.
If
1.
Then for any
8 > 0the path
hence letting
tend to
8
does not coincide with
easily obtain a strict inequality.
COROLLARY 10.7 .
Suppose that a path
by arclength,
the length of any
must con
0
The length of this the length of
r(Co,l]),
©
then we
This completes the proof of 10.1*.
An important consequence of Theorem 10.1*
metrized
©
5 to the spherical
and lying between these two shells.
©([0,1])
Consider any
to a point
where
shell of radius
=
±3 now straightforward.
is the following.
cd: [o,jU *■ M, para
haslength
less than or equal
other path from
©(0) tocd(j£) . Then
to ©
is a geodesic. PROOF:
Consider any segment of
above, and having length 10.1*.
< e.
Hence the entire path DEFINITION.
©
lying within an open set
W, as
This segment must be a geodesic by Theorem ©
A geodesic
is a geodesic. 7:
[a,b]
► M
will be called minimal
if
62
II. RIEMANNIAN GEOMETRY
its length is less than or equal to the length of any other piecewise smooth path joining its endpoints. Theorem 1 0 . 4 asserts that any sufficiently small segment of a geodesic is minimal.
On the other hand a long geodesic may not be minimal.
For example we will see shortly that a great circle arc on the unit sphere is a geodesic.
If such an arc has length greater than
*,
it is certainly
not minimal. In general, minimal geodesics are not unique.
For example two anti
podal points on a unit sphere are joined by infinitely many minimalgeodesics. However, the following assertion is true. Define the distance
p(p,q)
p,q e M
between two points
to be the
greatest lower bound for the arclengths of piecewise smooth paths joining these points.
This clearly makes
M
into a metric space.
It follows
easily from 10.4 that this metric is compatible with the usual topology of M. COROLLARY 10.8. Given a compact set K C M there exists a number 5 > 0 so that any two points of K with dis tance less than & are joined by a unique geodesic of length less than &. Furthermore this geodesic is minimal; and depends differentiably on its endpoints. PROOF.
Cover
K
by open sets Wa, as in 1 0 .3 , and let
small enough so that any two points in
K
with distance less than
5
be 5
lie
in a common Wa . This completes the proof. Recall that the manifold
M
is geodesically complete if every geo
desic segment can be extended indifinitely. THEOREM 10.9 (Hopf and Rinow*). If M is geodesically complete, then any two points can be joined by a minimal geodesic. PROOF. U
.r'
*
Given
p,q e M
as in Lemma 10.3. Let
S C U
with distance
r > 0,
choose a neighborhood
denote a spherical shell of radius
5 < e
Compare p. 341 of G. de Rham, Sur la r§ductibilite d ’un espace de Rlemann, Commentarii Math. Helvetici, Vol. 26 (1952); as well as H. Hopf and W. Rinow, Ueber den Begriff der•vollstandigen differentialgeometrischen Flache, Commentarii,Vol. 3 (1 9 3 1 ), pp. 209225.
§1 0 . GEODESICS AND COMPLETENESS about
p.
Since
S
is compact, there exists a point p0
on
S
63
I v 
=, expp (5v),
for which the distance to
q
=
1,
is minimized.
expp(rv)
=
q.
This implies that the geodesic segment
t ►
is actually a minimal geodesic from
to
p
We will prove that
7(t) = expp (tv),
0 < t < r,
q.
The proof will amount to showing that a point which moves along the
y
geodesic
must get closer and closer
to
q.
In fact for each
t € [5 ,r]
we will prove that O t)
p(r(t),q)
This identity, for
t = r,
=
rt
.
will complete the proof.
First we will show that the equality path from
p
to
q
must pass through
p(p,q) = Min
S,
(1 )
is true.
we have
(p(p,s) + p(s,q)) = 5 + p(p0 ,q)
s€S
p(pQ,q) = r  5.
Therefore
tQ € [s,r]
Let (1.)
is true.
z
If
tQ < r
point of
S'
.
0 Since
this proves (1 &).
pQ = 7(5),
denote thesupremum of those numbers
Then by continuity the equality
we will obtain a contradiction.
cal shell of radius
Since every
s!
p(r(tQ ),q) = Min
s €S '
(p(7 (t
0
for which
(1. ) is true also. S’
denote a small spheri
about the point
po €
with minimum distance from
0
Let
t
q.
a
(Compare Diagram 10.)
),s) + p(s,q)) = 5' + p(pi,q)
0
Then
,
hence (2)
p(Po,q) = (r  tQ)  5 * We claim that
p^
is equal to
.
7 ( ^ + 5').
In fact the triangle
inequality states that p (P,Pq )
(making use of (2)). p^
>
p (P^)
“
p (Pq
^)
= ^
+ 5’
But a path of length precisely
is obtained by following
a minimal geodesic from
7
y(tQ)
from to
p^.
p
to
y(tQ),
tQ + 5 f
from
p
and then following
Since this broken geodesic has
minimal length, it follows from Corollary 10.7 that it is an (unbroken)
to
64
II. REIMANNIAN GEOMETRY
geodesic, and hence coincides with Thus
7(tQ + 5 1) = p^.
7.
Now the equality (2) becomes
p(7 (^0 + 5 '),q) = r  (t0 + 8')
(ito+5l)
This contradicts the definition of
.
tQ; and completes the proof.
Diagram 10. As a consequence one has the following. COROLLARY 10.10. If M is geodesically complete then every bounded subset of M has compact closure. Con sequently M is complete as a metric space (i.e., every Cauchy sequence converges). PROOF. exppi
TMp*’M
If X C M
has diameter
maps the disk of radius
d d
then for any in TMp
of M which (making use of Theorem 10.9) contains of X
p €X
the map
onto a compact subset X.
Hence the closure
is compact. Conversely, if M
is complete as a metric space, then it is not
difficult, using Lemma 10.3, to prove that M
is geodesically complete.
For details the reader is referred to Hopf and Rinow.
Henceforth we will
not distinguish between geodesic completeness and metric completeness, but will refer simply to a complete Riemannian manifold.
§1 0 . GEODESICS AND COMPLETENESS FAMILIAR EXAMPLES OF GEODESICS. the usual coordinate system dx1 0 dx1 + ...+ dxn 0 dxR desic
7,
given by
X j ,...,xn
In Euclidean nspace,
Rn,
with
and the usual Riemannian metric
b
we have
65
= 0
t ► (x^ (t),...,xR (t)
and the equations for a geo
become
d 2x.
whose solutions are the straight lines. follows:
This could also have been seen as
it is easy to show that the formula for arc length
i1 coincides with the usual definition of aru length as the least upper bound of the lengths of inscribed polygons;
from this definition it is clear that
straight lines have minimal length, and are therefore geodesics. The geodesics on intersections of
PROOF.
Sn
are precisely the great circles, that is, the
with the planes through the center of
Reflection through a plane
whose fixed point set is with a unique geodesic
C = Sn (1 E 2 . C*
is an isometry, the curve between
Sn
I(x) = x
and
Let
E2 x
Sn .
is an isometry and
y
I(y) = y.
Then, since
is a geodesic of the same length as Therefore
Sn *■ Sn
be two points of
of minimal length between them. I(O')
I:
C' = I(C’).
C I
C*
This implies that
C 1 C C. Finally, since there, is a great circle through any point of
Sn
in
any given direction, these are all the geodesics. Antipodal points on the sphere have a continium of geodesics of minimal length between them.
All other pairs of points have a unique geo
desic of minimal length between them, but an infinite family of nonminimal geodesics, depending on how many times the geodesic goes around the sphere and in which direction it starts. By the same reasoning every meridian line on a surface of revolution is a geodesic. The geodesics on a right circular cylinder
Z
are the generating
lines, the circles cut by planes perpendicular to the generating lines, and
66
II.
the helices on
Z.
PROOF:
If
isometry
I:
The geodesics on in R
2
.
L
is a generating line of
 L ► R
Z
RIEMANNIAN GEOMETRY
2
by rolling
Z
onto
are just the images under
Two points on
Z
Z
then we can set up an 2
R :
I
of the straight lines
have infinitely many geodesics between them.
PART III THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS
§11.The Path Space of a Smooth Manifold. M
Let
be a smooth manifold and let
sarily distinct) points of will be meant a map 1) [0,1 ]
co:
cd(o )
[0,1 ] ► M
o> [t^_.j,t^]
* p
and
be
two (not neces
path
from p
to
q
0 = tQ < t1
*
dd x d d 'd t > = Therefore dE(a(u)) du
1 f / da 8a ^ J Xdd'dd > o
d = du
By Lemma 8 . 7 we cansubstitute Choose each strip
=
T\
jA
(e,e) x [t^_1,t^].
8 / 8a
in this last formula.
so that
Then we can
a
is differentiable on
"integrate by parts" on
The identity
8a v
/ D 8a
8a v
/ 8a D 8a x
dd X d u 'd t > = x d d du'db > + Xdu'dd dd > implies that t. f / D 8a 8a \ J X dd dCLdd > Li1
t=t. / 8a 8a v < dd'dd >
=
t. f /
"3
*
/y
for
0 = tQ < t1 at Hi
‘
72
III. CALCULUS OF VARIATIONS
Adding up the corresponding formulas for that
= °
for
1 dE(a(u)) I  Hu
k1 V / =
• Z
* J < H u ’ HI HI
1=1 Setting
i = l,...,k;
t = 0 or 1, this gives
1
...
> dt
•
o
a = 0, we now obtain the required formula 1
(o)
■
i < w >Atv >  a t 0
< w ’a >
dt
•
This completes the proof. Intuitively, the first term in the expression for that varying the path decrease
oi in thedirection of decreasing
E; see Diagram
— (o)
"kink,” tends
shows to
11.
Diagram 11 . The second term shows that varying the curve in the direction of its acceleration vector ^
(^)
tends to reduce
E.
Recall that the path oi e ft is called a geodesic oi
is C°° on the whole interval [o,1],
of
ai is identically zero along
if and only if
and the acceleration vector
oi.
COROLLARY 12.3. The path oi is a critical point for the function E if and only if oi is a geodesic.
§1 2 . THE ENERGY OF A PATH PROOF: critical point. f(t)
73
Clearly a geodesic is a critical point. There is a variation of
co with
is positive except that it vanishes at the
This is zero if and only if
=
where
t.. Then 1
< A(t),A(t)
0 A(t) = o
co be a
W(t) = f(t)A(t)
1
 ? H (0)
Let
for all
t.
> dt . Hence each
^[ti,ti + 1 3
is a geodesic. Now pick a variation such that W(t.) = A. V. Then V' i \ a§(0) =  2^ . If this is zero then all a^V co is differentiable of class
C1, even at the points
t^.
from the uniqueness theorem for differential equations that everywhere:
thus
co is an unbroken geodesic.
are
o,
and
Now it follows co is
C°°
71*
III.
CALCULUS OF VARIATIONS
§13 . The Hessian of the Energy Function at a Critical Path. Continuing with the analogy developed in the preceding section, we now wish to define a bilinear functional E**: Tnr x Tnr when
7
R
is a critical point of the function
E,
i.e., a geodesic.
bilinear functional will be called the Hessian of E If point
p,
f
at
is a real valued function on a manifold
This
7.
M
with critical
then the Hessian f**:
x T^
can be defined as follows. Given
 R
X1,X2 e TM^
choose a smooth map
(u1,u2) *• a(u1,u2) defined on a neighborhood of values in M,
(0,0) in R 2, with
so that
“ (0 , 0 )
=
 2 (0, 0)
P,
=
x,,
=
g2(°,o)
x2
.
Then f**(X,,X2)
3 2f (a(u, ,u„)) =2
1
This suggests defining E#*
as follows.
(0, 0)
Given vector fields W 1,W2
e TsU
choose a 2parameter variation a: where U
is a neighborhood of
a(o,o,t)
(Compare §11 .)
=
7(t),
U x [0,1 ] ► M , (0,0) in R 2, so that
"3 1 ^
=
Then the Hessian
^1
(^),
( ° > 0 >t )
=
W2 ( t )
E**(W.j,W2) will be defined to be the
second partial derivative 8 2E ( 5 ( u 1 ,u 2))
SU1 Su2
where a(u1,u2) € ft denotes the path
(0, 0)
a(u.,,u2)(t)
second derivative will be written briefly as
=a(u1,u2,t) (0,0)
The following theorem is needed to prove that
E**
. This
. is well defined.
§13.
THE HESSIAN OF
E
75
THEOREM 13. 1 (Second variation formula). Let a: U *■ft be a 2parameter variation of the geodesic 7 with variation vector fields wi

6 Tv 1 d2F i
Then the second derivative
1 " 1'2 • (0,0)
of the energy
function is equal to v 2
DW, c < W 2(t),At ^ 1 > 
D 2W. < V8,  4
J
77 z
+R(V,W1) V > dt
5
dt
0
where
V *^
denotes the velocity vector field and where DW.
DW.
DW.
At “err = ‘H T (t+) ■ *ur(t") DW.
denotes the jump in at one of its finitely many points of discontinuity in the open unit interval. PROOF:
According to 12.2 we have _
1 dE ^du2

J
“L ^
t dU > " J
du*
Y “
Zt ^/
8a
1
f 'J
D
da >.
dUg'dUj
At d^b >
1
/ \
D
0
da Dda^,. du ' db db > dt
1
Let us evaluate this expression for
f ’ J
d
D D da x dt. du.dU d b >
0
(u1 ,u2) = (0,0).
Since
7 = a(o,o) is
an unbroken geodesic, we have
A da
D
A t HI
=
°»
da
HI HI
_ =
0 ’
so that the first and third terms are zero. Rearranging the second term, we obtain
1
n2t
(1 3 2)
7 HG 1'Iu,2(0’0)
=
Y
* I