Handbook of Finsler Geometry: 2 Volume set [2, 1 ed.] 1402015569, 9781402015564

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TABLE OF CONTENTS

Preface Part 1

xv Complex FinslerGeometry

3

Tadashi Aikou 1 Kahler Fibrations...................................................................................................... 9 1.1 Fibrations ...................................................................................................... 9 1.2 Local Treatments............................................................................................ 10 1.3 Bott Connections............................................................................................13 1.4 Kahler Fibration ............................................................................................ 18 2 Complex Finsler Bundles 23 2.1 Vector Bundles Over Complex ProjectiveSpace...................... 23 2.2 Complex Fuidei Metrics............................................................................ 27 2.3 Bott CoBnectious of Fuislei VectorBundles ................................ 35 2.4 Negativity of Vector Bundles.................................................................. 39 2.5 Special Finsler Vector Bundles ............................... . . . 48 3 Kobayashi Metrics...........................................................................................................59 3.1 PoincareMetrics . . .... .................................................................................59 3.2 Kobayashi Metric........................................................................................... 62 3.3 Bounded Domains........................................................................................... 65 3.4 Holomorphic SectionalCurvature and Schwarz Lemrna . . 67 Part 2

KCC Theory of a System of Second Order Differential Equations

83

P.L. Antonelli and I. Bucalaru 1 TheGeometryoftheTangentBundle............................................................. 91 1.1 The Tangent Bundle...................................................................................... 91 1.2 The Vertical Subbundle ............................................................................ 93 1.3 The Almost Tangent Structure ............................................................. 94 1.4 Vertical and Complete Lifts .................................................................. 94 1.5 IIuiiiQgeneity............................................................ 95 2 Nonlinear Connections ................................................................................. 97 2.1 Horizonttil Distributions and Horizontal Lifts ... . 97 2.2 Characterizations of a Nonlinear Connection ............................... 99 2.3 Curvature and Torsion for a Nonlinear Connection 102 2.4 Aiitoparallel Curves and Symmetries for a Nonlinear Connection .................................................................. 103 2.5 Homogeneous Nonlinerir Connection............................... 107

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Anastasiei and Antonelli

3

4

5 6

7

Part 3

Finsler Connections on the Tangent Bundle ................................... 109 3.1 The Berwald Connection.................................................................. Ill 3.2 The h and v-Covariant Derivation of a Finsler Connection ........................................................................................... 112 3.3 The Torsion of a Finsler Connection......................................... 113 3.4 The Curvature of a FinslerConnection .. ..................................... 114 3.5 Finsler Connections Induced by a Complete Parallelism ................................................................................. 116 3.6 The Cartan Structure Equations of a Finsler Connection ........................................................................................... 118 3.7 , Geodesics of a Finsler Connection.............................................. 120 3.8 Homogeneous Berwald Connection ......................................... 121 Second Order Differential Equations........................................................ 123 4.1 Semispray or Second Order Differential Vector Field . . 123 4.2 Nonlinear Connections and Semisprays..................................... 125 4.3 The Berwald Connection of a Semispray.............................. 127 4.4 The Jacobi Equations of a Semispray ................................... 129 4.5 Symmetries for a Semispray ........................................................ 131 4.6 Geometric Invariants in KCC-Theory ................................... 132 Homogeneous Systems of Second Order Differential Equations......................................................................,...................................... 1.35 Time Dependent Systems of Second Order Differential Equations............................................................................................................... 139 6.1 Sprays and Nonlinear Connections on Jets......................... 139 6.2 Variational Equations ....................................................................... 144 6.3 The “Film-Space” Approach to Type (B) KCC-Theory . 147 The Classical Projective Geometry of Paths......................................... 151 7.1 Paths, Parametrized Paths..................................................... 151 7.2 The Various Geometries of Paths- Finite Equations . . 152 7.3 The Various Geometries of Paths - DifferentialEquations 153 7.4 Affine Connections ........................................ 155 7.5 TheFundamentaiprojectiveInvariants .................................... 158 7.6 The Projective Parameter and the Normal Spray Connection...................................................... 161 7.7 Projective Deviation..................................................................... 165 Fundamentals of Finslerian Diffusion with Applications

177

P.L. Antonelli and T.J. Zastawniak 1 FinslerSpaces ..................................................................................................... 1.1 TheTangentandCotangentBuiidle .................................... 1.2 Fiber Bundles ...................................................................................... 1.3 Frame Bundles and Linear Connections.............................. 1.4 Tensor Fields ...................................................................................... 1.5 Linear Connections............................................................................ 1.6 TorsionandcurvatureofaLiiiearConnection ....

187 167 189 191 192 19-1 196

Table of Contents

vii

1.7 Parallelism ............................................................................................ 197 1.8 The Levi-Civita Connection on a Riemannian Manifold 197 1.9 Geodesics, Stability and the Orthonormal Frame Bundle 199 1.10 Finsler Space and Metric ............................................................. 200 1.11 FinslerTensorFields................................................................ 202 1.12 Nonlinear Connections .................................................................. 202 1.13 Affine Connections on the Finsler Bundle.................. 204 1.14 Finsler Connections ....................................................................... 206 1.15 TorsionsandcurvaturesofaFinslerConnection . . . 208 1.16 Metrical Fihsler Connections. The Cartan Connection . 210 2 Introduction to Stochastic Calculus on Manifolds.......................... 213 2.1 Preliminaries............................................................................................ 213 2.2 Ito’s Stochastic Integral .................................................................. 216 2.3 Ito’s Processes. Ito Formula ........................................................ 219 2.4 Stratonovich Integrals....................................................................... 221 2.5 Stochastic Differential Equations on Manifolds .... 221 3 Stochastic Development on Finsler Spaces......................................... 227 3.1 Riemannian Stochastic Development......................................... 227 3.2 Rolling Finsler Manifolds Along Smooth Curves and Diffusions....................................................................................... 233 3.3 Finslerian Stochastic Development ......................................... 242 3.4 Radial Behaviour................................................................................. 246 4 Volterra-Hamilton Systems of Finsler Type......................................... 249 4.1 Berwald Conjiections and Berwald Spaces.......................... 249 4.2 Volterra-Hamilton Systems and Ecology............................... 253 4.3 Wagnerian Geometry and Volterra-Hamilton Systems . 254 4.4 Random Perturbations of Finslerian Volterra-HamiltonSystems ........................................................ 260 1.5 Random Perturbations of Riemannian Volterra-Hamilton Systems ........................................................ 262 4.6 Noise in Conformally Minkowski Systems .......................... 266 4.7 Canalization of Growth and Development with Noise 267 4.8 Noisy Systems in Chemical Ecology and Epidemiology 271 4.9 Riemannian Nonlinear Filtering................................................... 279 4.10 ConformaisignalsandGeometryofFilters ..................... 285 4.11 Riemannian Filtering of Starfish Predation ..................... 289 5 Finslerian Diffusion, and Curvature ........................................................ 295 5.1 Cartan’s Lemma in Berwald Spaces......................................... 296 5.2 Quadratic Dispersion ....................................................................... 29δ 5.3 Finslerian Development and Curvature.................................... 299 5.4 Finslerian Filtering and Quadratic Dispersion.................... 300 5.5 Entropy Production and Quadratic Dispersion .... 302 6 Diffusion on the Tangent and Indicatrix Bundles .......................... 319 6.1 Slit Tangent Bundle as Riemannian Manifold.................... 320 6.2 Au-Development as Riemannian Development with Drift 321 6.3 Indicatrized Finslerian Stochastic Development .... 323

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Anastasiei and. Antonelli

6.4 Indicatrized /^-Development Viewed as Riemannian . . 327 Appendix A Diffusion and Laplacian on the Base Space .... 335 A.l Finslerian Isotropic Transport Process.................................... 336 A.2 Central Limit Theorem .................................................................. 338 A. 3 Laplacian, Harmonic Forms and Hodge Decomposition 340 Appendix B Two-Dimensional Constant Berwald Spaces .... 343 B. l BerwakFs Famous Theorem ........................................................ 343 B.2 Standard Coordinate Representation ............................................ 344 B.3 B2(I) with Constant G⅛k .............................................................. 345 B.4 Class B2(2) with Constant Gjk..................................................... 347 B.5 - B2(r,s) with Constant G^k .......................................................... 348 Part 4

Symplectic Transformation of the Geometry of T*M; L-Duality

D. Hrimiuc and H. Shimada 1 The Geometry of TM and T*M ............................................................... 1.1 Connections on TM .......................................................................... 1.2 Semisprays and Connections........................................................ 1.3 Linear Connections on TM .......................................................... 1.4 The Geometry of Cotangent Bundle .................................... 1.5 LinearConnectionsonT+M . . . . ....................................... 1.6 Lagrange Manifolds............................................................................ 1.7 Hamilton Manifolds............................................. 2 Symplectic Transformations of the Differential Geometry of T* M........................................................................................................................ 2.1 Connection-Pairs on Cotangent Bundle ............................... 2.2 Special Linear Connections on T*M ..................................... 2.3 The Homogeneous Case.................................................................. 2.4 /-Related Connection-Pairs ........................................................ 2.5 /-Related ^-Connections ............................................................. 2.6 The Geometry of a Homogeneous Contact Transformation ............................................................. 2.7 Examples................................................................................................. 3 The Duality Between Lagrange and Hamilton Spaces .... 3.1 The Lagrange-Hamilton L-Duality ......................................... 3.2 L-Dual Nonlinear Connections................................................... 3.3 L-Dual d-Connections....................................................................... 3.4 The Finsler-Cartan L-Duality ................................................... 3.5 Berwald Connection for Cartnn Spaces. Landsberg and Berwald Spaces. Locally Minkowski Spaces . . 3.6 Applications of the L-Duality ...................................................

359

363 363 368 370 373 376 378 381 385 385 390 395 398 403 405 409 413 413 417 421 426

431 435

Table of Contents

Part 5

Holonomy Structures in Finsler Geometry

L. Kozma 1 Holononiy of Positively Homogeneous Connections ..................... 1.1 Connections of a Tangent Bundle.............................................. 1.2 Holonomy Group of a Positively Homogeneous Connection 1.3 Curvature and Holonomy Algebra of a Positively Homogeneous Connection............................................................. 1.4 Homogeneous Holonomy of Finsler Manifolds .... 1.5 Metrizability of Positively Homogeneous Connections . 2 Holonomies of Finsler Vr-Connections ................................................... 2.1 A Topological Group and Its Lie Algebra .......................... 2.2 Vr-Connections...................................................................................... 2.3 The Vr-Holonomy Group and Vr-Holonoiny Algebra . . 3 Holonomies of the Finsler Vector Bundle.............................................. 3.1 Linear Connections of the Finsler VectorBundle . . . 3.2 Osculation of Finsler Pair Connections ......................... 3.3 Ziy-Holonomy Groups of the Finsler Vector Bundle . . 3.4 Tho Mixed Holonomy Groups ................................................... 4 Holonomies of Special Finsler Manifolds.............................................. 4.1 Berwald Manifolds ............................................................................ 4.2 Landsberg Manifolds ....................................................................... Part 6

On the Gauss-Bonnet-Chern Theorem in Finsler Geometry

ix

445

453 453 454

455 458 458 463 463 464 465 469 469 470 472 473 477 477 481

491

Brad Lackey 1 Topological Preliminary.................................................................................. 2 The Method of Transgression....................................................................... 3 The Correction Term ....................................................................................... 4 Special Cases........................................................................................................... 4.1 Riemannian Geometry .................................................................. 4.2 The Chern Connection .................................................................. 4.3 A Special Family of Finsler Connections...............................

497 499 503 505 505 505 506

The Hodge Theory of Finsler-type Geometries

513

Brad Lackey 1 Elliptic Complexes ............................................................................................ 1.1 The Hodge-deRham Complex ................................................... 1.2 Elliptic Complexes ............................................................................ 1.3 Elliptic Operators ............................................................................. 1.4 The Hodge Decomposition Theorem .................................... 2 The Weitzenbock Formula............................................................................. 2.1 Complete Positivity............................................................................ 2.2 Covariant Formalism ....................................................................... 2.3 Existence and Uniqueness of a Connection......................... 2.4 A Bochner Vanishing Theorem...................................................

521 521 523 527 531 533 534 536 539 541

Part 7

Anastasiei and Antonelli

X

3

Part 8

Complete Positivity of the Symbol ....................................................... 3.1 The Geometric Ratio....................................................................... 3.2 Computing the Geometric Ratio .............................................. 3.3 An Example........................................................................................... Finsler Geometry in the 20th-Century

543 543 545 547 557

M. Matsumoto 1 Finsler Metrics..................................................................................................... 565 1.1 Extremals................................................................................................ 565 1.2 Finsler Metric ...................................................................................... 569 1.3 .RandersMetric...................................................................................... 574 1.4 (α, β)-Metric........................................................................................... 581 1.5 I-Form Metric...................................................................................... 587 1.6 m-th Root Metric ............................................................................ 592 1.7 Birth of Finsler Geometry............................................................. 595 2 Connections in Finsler Spaces .................................................................. 601 2.1 Frame Bundles...................................................................................... 601 2.2 Linear Connections............................................................................ 607 2.3 Vectorial PYame Bundles ............................................................. 618 2.4 The Theory of Pair Connections .............................................. 628 2.5 Standard Finsler Connections ................................................... 644 2.6 Special Finsler Connections ........................................................ 661 3 Important Finsler Spaces ............................................................................. 677 3.1 Finsler Space of Dimension 'Γwo .............................................. 677 3.2 Riemannian Space and Locally Minkowski Space . . . 709 3.3 Stretch Curvature and Landsberg Space............................... 717 3.4 Berwald Space..............................................................................................723 3.5 Wagner Space............................................................................ 735 3.6 Scalar Curvature and Constant Curvature.......................... 741 3.7 Finsler Space of Dimension Three ......................................... 753 3.8 Indicatrix and Homogeneous Extension............................... 775 4 Conformal and ProjectiveChange ......................................................... 783 4.1 Conformal Change ............................................................................ 783 4.2 Conformally Flat Finsler Space................................................... 790 4.3 Conformal Change and Wagner Space.................................... 796 4.4 Projective Change ............................................................................ 799 4.5 Douglas Space...................................................................................... 814 4.6 Finsler Space with Rectilinear Extremals .......................... 827 5 Finsler Spaces with I-Form Metric and with m-th Root Metric 839 5.1 Finsler Spaces with I-Form Metric................................................ 839 5.2 Curvature of Two-Dimensional I-Form Metric .... 847 5.3 ConformalChangeofl-FormMetric .................................... 851 5.4 Finsler Space with m-th Root Metric.................................... 858 5.5 Stronger Non-Riemannian FinsIer Space............................... 867 5.6 Two-Dimensional m-th Root Metrics .................................... 874 5.7 Berwald Spaces of Cubic and Quartic Metrics .... 879

Table of Contents

6

Part 9

xi

Finsler Spaces with (a, ^-Metrics ........................................................ 6.1 Fundamental Tensor of Space with (α,∕3)-Metric . . . 6.2 C-Tensors of (α, β)-Metrics ........................................................ 6.3 Connections for (α, /J)-Metrics................................................... 6.4 Douglas Space with (a,/J)-Metric.............................................. 6.5 Two-Dimensional Space with (α∕3)-Metric.......................... 6.6 Strongly Non-Riemannian (α∕3)-Metric ............................... 6.7 Conformal Change of (α, β)-Metric......................................... 6.8 Projective Change of (αβ)-Metric.............................................. 6.9 Randers Spaces of Constant Curvature ............................... The Geometry of Lagrange Spaces

889 889 894 901 913 916 924 928 936 946 969

Radu Miron, Mihai Anastasiei and loan Bucataru 0 Introduction.......................................................................................................... 973 1 The Geometry of theTangentBundle....................................................... 977 1.1 The Manifold TM ............................................................................... 977 1.2 Semisprays on theManifoldTM................................................... 984 1.3 Nonlinear Connections .................................................................. 987 1.4 A-Linear Connections....................................................................... 995 1.5 Semisprays, Nonlinear Connections and TV-Linear Connections.................................................................................................1002 1.6 Parallelism. Structure Equations................................................... 1007 2 Lagrange Spaces ..................................................................................................... 1013 2.1 TheNotionofLagrangeSpace........................................................ 1013 2.2 Geometric Objects Induced on TM by a Lagrange Space 1017 2.3 Variational Problem and Euler-Lagrange Equations . . 1019 2.4 A Noether Theorem ............................................................................ 1021 2.5 Canonical Semispray. Nonlinear Connection..........................1023 2.6 Geodesics in a Finsler Space............................................................. 1025 2.7 Hamilton-Jacobi Equations ............................................................. 1028 2.8 The Almost Kahlerian Model of a Lagrange Space Ln . 1030 2.9 Metrical Ar-Linear Connections........................................................ 1033 2.10 Almost Finslerian Lagrange Spaces.............................................. 1038 2.11 Geometry of φ-Lagrangians ............................................................. 1042 2.12 Gravitational and Electromagnetic Fields...............................1045 2.13 Einstein Equations of Lagrange Spaces .................................... 1047 3 Subspaces in Lagrange Spaces ....................................................................... 1053 vτn

3.1 3.2 3.3 3.4 3.5 3.6 3.7

Subspaces L in a Lagrange SpaceLn .......................................1053 Induced Nonlinear Connection........................................................ 1056 The Gauss-Weingarten Formulae................................................... 1060 The Gauss-Codazzi Equations ........................................................ 1061 Totally Geodesic Subspaces ............................................................. 1062 Lagrange Subspace of CodimensionOne...................................... 1064 Subspaces in Finsler Spaces ............................................................. 1067

xii

Anastasiei and Antonelli 4

5

Part 10

Generalized Lagrange Spaces............................................................................ 1073 4.1 The Notion of Generalized Lagrange Space .......................... 1074 4.2 Metrical Ar-Connection in a GfL-Space ....................................1077 4.3 GfL-Metrics Determining Nonlinear Connections . . . 1080 4.4 GL-Metrics Provided by Deformations of Finsler Metrics............................................................................................1085 4.5 Almost Hermitian Model of a Generalized Lagrange Space ...................................................................................... 1091 Rheonomic Lagrange Geometry....................................................................... 1097 5.1 Semisprays on the Manifold TM × R.......................................... 1097 5.2 Nonlinear Connections on E = TM × R..................................... 1099 5.3* VariationalProblem ........................................................................... IlOl 5.4 RheonomicLagrangeSpaces............................................................. 1103 5.5 Canonical Nonlinear Connection ................................................... 1104 5.6 An Almost Contact Structure on E............................................... 1105 5.7 AT-Linear Connection............................................................................ 1107 5.8 Parallelism. Structure Equations for AT-Linear Connections............................................................................ 1108 5.9 Metrical AT-Linear Connection of a Rheonomic LagrangeSpace ...................................................................................... Illl 5.10 Rheonomic Finsler Spaces.................................................................. 1112 5.11 ExamplesofTimeDependentLagrangians............................... 1114 Symbolic Finsler Geometry

1125

S. F. Rutz and R. Portugal 1 Computer Algebra for Finsler Geometry...................................................1129 1.1 Introduction.................................................................................................1129 1.2 Computer Algebra ................................................................................. 1130 1.3 ManipulationoflndicesviaGroupTheory............................... 1144 1.4 FINSLERPackage ................................................................................. 1150 Part 11

A Setting for Spray and Finsler Geometry

1183

Jozsef Szilasi 0 Introduction................................................................................................................ 1187 1 The Background: Vector Bundles and Differential Operators ...................................................................................... 1191 A Manifolds.......................................................................................................... 3191 B Vector Bundles...................................................................................... 1195 C SectionsofVectorBundles ..................................................................1204 D Tangent Bundle and Tensor Fields.................................................. 1208 E Differential Forms ......................................................................................1218 F Covariant Derivatives................................................................................ 1226 2 Calculus of Vector-Valued Forms and Forms Along the Tangent Bundle Projection................................................................................. 1237 A Vertical Bundle to a Vector Bundle.............................................. 1237 B Nonlinear Connections in a Vector Bundle.................................... 1245

Table of Contents

xiii

C Tensors Along the Tangent Bundle Projection. Lifts . . 1258 D The Theory of A. Frolicher and A. Nijenhuis...............................1272 E The Theory of E. Martmez. J. F. Carinena and W. Sarlet 1298 F Covariant Derivative Operators Along the Tangent Bundle Projection .......................................................................................1314 3 Applications to Second-Order Vector Fields and Finsler Metrics ......................................................................................................1347 A Horizontal Maps Generated by Second-Order Vector Fields 1347 B Linearization of Second-Order Vector Fields...............................1362 C Second-Order Vector Fields Generated by Finsler Metrics 1369 D Covariant Derivative Operators on a Finsler Manifold . . 1383 Appendix............................................................................................................................... 1399 A.l Basic Conventions................................................................................ 1399 A. 2 Topology.....................................................................................................1400 A.3 The Euclidean n-Space Rzl ............................................................ 1401 A.4 Smoothness................................................................................................1402 A.5 Modules and Exact Sequences....................................................... 1403 A.6 Algebras and Derivations................................................................. 1408 A.7 Graded Algebras and Derivations .............................................1409 A.8 Tensor Algebras Over a Module ..................................................1411 A.9 The Exterior Algebra........................................................................... 1415 A. 10 Categories and Functors .................................................................. 1419

Matsumoto

728

3.4.3

//-Curvature Dependent on Position Alone

Wc shall recall the A-Curvature tensors of the Berwald, Cartan and Chern-Rund connections. These are given by BΓ :

H^k = ‰G> - 0rG⅛)GJ + G^jG^k - (⅛),

GΓ:

^fc=Λ¾fc + C⅛,

CΛΓ :

K^k = ∂kF^ - (∂rF∣,j)Gk + F[jF * k - (⅛).

If we are concerned with a Berwald space, the Definition 3.4.1.1 and The­ orem 3.4.1.2 show that both G¾ and F-li. and hence the A-curvature tensors H and K are functions of position (xτ) alone, but R is not so, because C¾. may contain (τ∕t). Definition 3.4.3.1. The sets of n-dimensional Finsler spaces with the

Λ-curvature tensors Hy R and K dependent on position alone are denoted by

Hx(n)y

Rx(n)

and

Kx(n)

respectively.

If we denote further by B(n) the set of all n-dimensional Berwald space, that is, B(x) = {n-dim. Berwald spaces}, then we have the inclusion relations as follows: B(n) C Hx(n)y

B(n) C Kx(n).

(3.4.3.i)

Since (2.5.4.3) and (2.5.5.7) show yiK^jk=I⅛,

H⅛k = ∂iR⅛,

Fn ∈ Kx(ri) is also Fn ∈ Hx(n). Consequently, we have the inclusion relation Kx(n) C Hx(ri).

(3.4.3.2)

Next we introduce the sets: L(n) = {7i-dim. Landsberg spaces},

5(n) = {n-dim. spaces with zero stretch curvature tensor}. Then Propositions 3.3.3.1 and 3.4.1.1 lead to the inclusion relations B(n) C L(rι) C S(n). Theorem 3.4.3.1. We have the inclusion illations

B(n) C Kx(n) C Hx(ri) C S(n).

(3.4.3.3)

Finsler Geometry in the 20th-Century

729

Proof: It is sufficient to show the last relation: (3.3.2.5) can be written as

∑hO7. = yrH∣ιljk

Thus H — H(x) implies Σ — 0.

This theorem and (3.4.3.3) give rise to the problem: What are the intersec­ tions L(n) ∩ Hx(τι)t? L(n) ∩ Kx(n)t? Theorem 3.4.3.2. L(n) ∩ Kx(ri) = L(ri) ∩ Hx(n). Proof: We recall (2.5.5.2):

* G⅛=J + J¾. Theorem 3.3.3.1 shows G⅛ = F⅛ for Fn ∈ L(n). We have Rx(ri) C Hx(n) similarly to the case of Kx(n)i but Rx(ri) will be studied later on. Now we are especially interested in the twτo-dimensional case. The spaces we shall consider here are all contained in 5(2). On account of (3.3.2.6) and the Bianchi identity (3.1.3.14), such a space is characterized by Ia

,ι — ^R)2 +RI = 0.

Putting G-cj = liemj — Ijtnii (3.1.3.15) and (3.1.3.17) are written as Hfzhij

= (εRGkh + εRi2 mkm∣l)Giji

R-khij

= (εRGkh

RIτ∏ktrih)Gij.

Then (3.4.3.4) yields H = K for Fn ∈ 5(2). Consequently, Theorem 3.4.3.3. In the two-dimensional case

Hx(2) = Kx(2). According to Theorem 3.4.2.1 we have the direct sum expression

B(2) = Bi(2) + B2(2) + B3(2),

Bi(2) = {B — 0, I ≠ const.}, B2(2) = {R = 0, I = const.} B3(2) = {R ≠ 0. I = const.}. Let us now, on the other hand, consider Hx(2). Since we have from (3.2.3.15)

LH⅛k.( = ε{(Bj2 ;2 +εIR-i2 )mh - 2(R∖2 +εIR)In}mimtGjk∙

Matsumoto

730

Hence F2 ∈ Hx (2) is characterized by )2 ÷εIRι2 = 0?

R⅛^∣"εIR = 0,

but the latter has been shown in (3.4.3.4). The former gives I;2 R = 0. There­ fore, Theorem 3.4.3.4. A two-dimensional Finsler space belongs to Hx(2)i if and only if Iij = 0 and I;2 = 0.

Hx(2) is expressed by the direct sum Hx(2) = Hi(2) + II2(2) + H3 (2),

H1 (2)

= {H = 0, I52≠O, I1j = 0},

H2(2)

= {R = 0, I.2 = 0, Ijj = 0},

H3(2)

={Λ≠0, I52 = 0, Ijj= 0}.

Since F2 ∈ B(2) is characterized by Ij = I2 = 0, the above leads to Corollary 3.4.3.1. Bi(2) = B(2) ∩Hi(2), i = 1,2,3.

In other words, the well-known classification of B(2) (Theorem 3.4.2.1) can be induced from the classification of Hx('2) (Theorem 3.4.3.4). We consider the intersection L(2) ∩ Hx(2). From the character Ij = 0 of L(2) (Proposition 3.3.3.2) we get

L(2) ∩ H1 (2)

= {R = 0, I52 ≠ 0, I1 = 0},

L(2)∩⅞(2)

={H = 0, Ij2 = Ij =0},

L(2)∩H3(2)

= {R>0, I2 = Ij =0}.

From the Ricci identity (3.1.3.10, b), I;2 = Ij = 0 yield I2 = 0, that is, I = const. Therefore we get Theorem 3.4.3.5.

L(2)∩H1(2) DB1(2),

L(2)∩Hi(2) = Bi(2),

i = 2,3.

Now wre deal with the set Rx(n). We have the Bianchi identity (2.5.2.4, c). Expressing R[lj∖k in terms of (∙⅛), the identity can be written in the form R⅛-k + ¾hfcr¾∙ + ¾∕∙⅛ - ∕⅛vC‰ + ‰ = 0,

Qtkij = -¾j] {Ptjk1

- tnm1')Gijmk.

Thus Q =■ 0 is equivalent to 7,ι ,ι = 0, that is, the stretch curvature Σ = 0, from (3.3.2.6).

In the two-dimensional case, Q = 0 is equivalent to l,ι ?i = 0? that is, F2 ∈ S(2).

Proposition 3.4.3.2.

Thus all F2 ∈ 5(2) have Q = 0. On the other hand, the condition (3.4.3.4) can be written as

εlRinkfJjmh + ehm1)Gij = 0, with implies RI = 0. Therefore, Theorem 3.4.3.7. If a two-dimensional Finsler space with non-zero scalar

curvature R belongs to Rx(2), then it is a Riemannian space.

Thus a non-Riemannian F2 ∈ Rτ(2) has R = 0. and hence the inclusion B(2) C Rx(T) is false, so that Bfn) C Rxfri) will be not true. Ref S. Bacso and M. Matsumoto [16], [17]. In their papers ([15] they were greatly surprised and delighted at the discovery of the following remarkable fact: For a Douglas space the components W∙jk of the projective Weyl curvature tensor are functions of position alone. This fact enabled them to consider the theory of the present section. See Theorem 4.5.2.4.

732

3.4.4

Matsuinoto

C-Reducibility

We are concerned v√ith the C-tensor given by (1.2.2.5) of a Finsler space Fn. The vanishing of the C-tensor characterizes Riemannian space. Further, in any two-dimensional Finsler space the C-tensor is written in the simple form (3.11.10). Now we shall propose a simple form of the C-tensor. We must pay attention to the fact that Cijk is symmetric and satisfies Cijkyk = 0∙ The angular metric tensor hij is also symmetric and satisfies h[jyj = 0. Hence we may put Cjjk = di,hjk -J- djhki -∣- dkhij->

with some contravariant vector ⅛. By multiplying by y∖ we get ⅛yl = 0. Next, multiplying by gli, we get Ck = (n + 1)⅛. Consequently we have the form

_ (Cihjk + Cjhki + Ckh,j) Cijk ~ (^+i) ∙

. t3∙4∙4∙υ

It is remarked that the C-tensor of any two-dimensional case is of the form (3.4.4.1), because (3.1.1.10) and (3.1.1.2) enable us to write (3.4.4.1). Definition 3.4.4.1. A Finsler space Fn of dimension n ≥ 3 is called C-reducible,

if the C-tensor is of the form (3.4.4.1). We deal with a C-reducible Finsler space Fn. First we examine the identity Chij∖k - Chik∖j = 0. See (3.1.3.12). (3.4.4.1) leads to (n + l)ChtjIfc = S(jHj){Ch∖khij - hhk

If we construct the contracted T-tensor

Tij = Tijhkghk = LC⅛ + Ciej + Cjti,

(3.4.4.2)

then we have Chij∖k — Chik∖j = 0 in the form

h-ijThk -∣- hjhTik ~ hikThj — hhkTij — 0.

Multiplying by ghk and paying attention to 7⅛ = 0, we get Tij = Thij /(n - 1) with T = Tijg⅛ = LCr∣r. Therefore, Chij ∣fc

{h,ħkhij}

= { fc(n2 -1)

(thCijk+tiCιljk+tjC-rιli.,+tkChιj) L

Consequently, on account of the definition of the T-tensor, we have Thijk = ∣ -^2 Z j j

(3.4 4.3)

Finsler Geometry in the 20th-Century

733

Next we deal with the Λ∙-covariant derivative Chij,k with respect to the Cartan connection CT :

⅛=¾i)¾⅛1.

(3.4.4.4)

Then (2.5.2.14) leads to

∏

_ (hhiCjlo + hijCfl,Q + hjhCuq)

p*' ij -

------------------- ∙

(3∙4∙4∙5)

Further, from S⅛k = CfkC‰ - CfjC⅛s and (3.4.4.1) we get

qh bHk

τ

{hikC^ + h^Cik}

aW

(n + 1)2

(3.4.4.6) Ci} = (τ)fty + CiCj'

°2 = giiCiCj-

Proposition 3.4.4.1. Let Fn, n ≥ 3, be a C-reducible Finsler space. The T-tensor, the (y)hυ-torsion tensor and v-curvature tensor of the Cartan con­ nection CΓ of Fn are written as (3.4.4.3), (3.4.4.5) and (3.4.4.6) respectively.

Now, suppose that Fn, n ≥ 3, is a Landsberg and C-reducible Finsler space. Then (2.5.2.14, b) is reduced to Cjkiih ~ Cjkhii = 0,

from which we have C41/,. - C∕llj = 0. Substituting from (3.4.4.4) in the above and multiplying by ghk, we get immediately Ci,j-μhij, We have the Bianchi identity (2.5.2.11, b). For the Landsberg space it is reduced to Sijkif = θ∙ Fpom (3.4.4.6) this is written as follows: First we get Cijlk = (n + F)μCijk∙ Therefore S⅛kιe = 0 yields

μ(h>kChjf ÷ hfljCikf

⅛ijChkt

hhkCijf^ = 0.

Multiplying by gikghj, we get 2μ(n - 2)Cf = 0. If μ = 0, then Cilj = 0, so that Cflfjik ~ 0 and hence Fn is a Berwald space from Theorem 3.4.1.3. Next, Cf =- 0 gives rise to Crijfc = 0 from (3.4.4.1). Therefore we obtain Theorem 3.4.4.1. If a Landsberg space Fn, n ≥ 3, is C-reducible, then Fn is

a Berwald space. Ref. Theorem 3.4.4.1, shown by M. Matsumoto [86], was the first of the Reduction Theorems of Landsberg spaces. See the end of §3.4.2.

Matsunioto

734

Let Fn, n ≥ 3, be a C-reducible Finsler space such that BΓ and CΓ of Fn have the same ∕ι-curvature tensors. Then (2.5.5.13) with (2.5.2.14,a) gives Gιkr∣0^¾l0 -

ChjrιθC⅛klQ -■ 0.

From (3.4.4.1) it follows that the above can be written as

^[jk]{Cr∣oC^hhk^ij + ChloCiilohjj + GiθCjlo⅛fc} = 0.

Multiplying by hhk, the above gives (n

l)Cr∣oClθ∕ι⅜j

(n

3)C∖∣oCj∣o = θ∙

From rank (hij) = n—1 and the assumption n ≥ 3 it follows that the above yields CrιoCjo = 0, and, if n ≥ 4, then we get C⅛lo = 0; and hence C∕l∣√lo = Phij = 0 from (3.4.4.5). If n = 3 and the metric is positive-definite, then we get Cr.o — 0. In every case, Theorem 3.4.4.1 shows that Fn is a Berwald space. Therefore we conclude Theorem 3.4.4.2. If α C-reducible Finsler space Fn, n ≥ 3, has the common

h-curvature tensors of the Berwald and Cartan connections, then and

Ct,qC^ =

0

(1) n ≥ 4 : Fn is a Berwald space, (2) n = 3 : Fn is a Berwald space, provided that the metric is positive-definite. Corollary 3.4.4.1. If a C-reducible Finslcr space Fr∖ n ≥ 3, has vanishing h-curvature tensor of the Cartan connection, then

(1) ∕ι ≥ 4 : Fn is a locally Minkowski space, (2) n = 3 : Fn is a locally Minkowski space, provided that the metric is positivedefinite.

Proof: We have the identities

yhFihjk = R * k,

OhRjk = ^hjkf

from (2.5.2.5) and (2.5.5.7). Hence Rlhjk = 0 implies Hkjk = 0. Thus Corollary 3.4.4.1 is a special case of Theorem 3.4.4.2. See Theorem 3.2.4.2. Remark: See §6.2.3 where the existence theorem of C-reducible Finsier spaces

are established.

Finsler Geometry in the 20th-Century

3.5

735

Wagner Space

3.5.1 Generalized Berwald Space Let us recall the Finsler connection which was given by Theorem 2.6.6.1. There, the four axioms except (2) are common with those of Definition 2.5.2.1 of the Cartan connection. Definition 3.5.1.1. We have a Finsler connection which is uniquely determined from the fundamental function L(x, y) and a skew-symmetric tensor field T of (l,2)-type of the system of five axioms:

(1) h-metrical,

(2) (h)h-torsion T, (3) deflection tensor D = 0, (4) v-metrical, (5) (v)v-torsion S1 = 0. This connection is called a generalized Cartan connection with the torsion T and is denoted by CT(T) = (F]kiN^ ¾). Cjk arc components of the C-tensor, which are given by (4) and (5). Here we suppose that T is (0)p-homogeneous as usual. The symbols (l, ∣) are used to denote the h and v-covariant differentiations in CT(T). As has been shown in Theorem 3.4.1.2, a Berwald space is characterized by Fjk = F * k(x) of CT. Generalizing this notion to CT(T), we propose Definition 3.5.1.2. A Finsler space with a skew-symmetric tensor T of (1,2)type is called a generalized Berwald space (with respect to T), if the connection coefficients Fjk of CT(T) are functions of position alone.

Now we are concerned with a generalized Berwald space Fn. On account of (2.4.3.3), Fn is characterized by Phkij = -Chkjn + ChkrPij-

(3.5.1.1)

Since CΓ(T) is h and v-metrical, we have Phkij + Pkhij = θ from the Ricci identity (b) of (2.4.3.8). Thus the left-hand side of (3.5.1.1) must be skewsymmetric in (h, A:), while the right-hand side is obviously symmetric in (h,k). Hence we have Phkij = 0 and Chkj,i = ChkrPjj- Since CT(T) satisfies the D and [/-conditions, Theorem 2.4.5.1 leads to Pijk = Poijk = θ∙ Therefore (3.5.1.1) is reduced to Phkij = θ> Chkjvi = 0. (3.5.1.2) Under the conditions (3.5.1.2) the Bianchi identity (a) of (2.4.4.4) is reduced to

τ⅛∖k - C^kτrj + τ C⅛ *

- TjlrC[k = O1

736

Matsunioto

which is nothing but Th,k = 0, that is, T∕j∙ are functions of position alone. Conversely, if Chkjii = 0 and T∙tj.k = 0, then (2.4.4.4,a) yields

∙z⅛j] {pihτPjk ~ Pihjk} = θ∙ By the Christoffel process with respect to (∕ι,i,J), the above leads to

Phijk ÷ CijrPrhk — CtijrPrk = 0. Multiplying by yh and next by y∙i, the above gives Pijk 4" CijrPok = 0,

PiQk = θι

which implies Phijk = 0∙ Therefore we return to the necessary and sufficient conditions (3.5.1.2). Hence we obtain the following theorem quite similar to Theorem 3.4.1.3. Theorem 3.5.1.1. A Finsler space with a generalized Cartan connection CT(T)

is a generalized Berwald space, if and only if the components of T are functions of position alone and Chijik = 0. Ref. V. Wagner [167]. The exact formulation of the notion of generalized Berwald space was given by M. Hashiguchi [46].

For the later use, we shall find the difference of CT(T) with the Cartan con­ nection CT. Here we denote GT = (FLCTC‰) and CT(T) = (*Fk,7Vj, C‰) and put *ηk = ηk + D⅛. (3.5.ι.3) Since CT(T) has the vanishing deflection tensor, (3.5.1.3) implies

Nl=Gik + D'0k. The condition gijlk = 0 in CT(T) yields

Dijk 4~ Djik + 2C}jDork = 0,

Dijk = 9jrD^k.

Applying the Christoffel process to the above and paying attention to Djik — Dkij = Tjik (= ffirTfk), we obtain Dijk = Atjk ~∙ CijD()rk — CjkDθri 4^ CkiDθrj>

2Atjk == Tijk ~ Tjki 4" Tkij. Multiplying by yl and next by yk, the above yields

Dojk = Aojk - CrkDorQ.

A)jO =

JO ∙

Consequently we obtain

Dijk = Aijk — Cij(Aork — Cyk-Aθsθ) ~ Cjk(Aθri — C8 riAθsθ) (3.5.1.4) ÷ Cfci(A)rj - C¾Aθsθ)∙

Finsler Geometry in the 20th-Century

737

Proposition 3.5.1.1. The difference Djk = * Fjk-Fjk of CT(T) = (* Fj k1 Nj1 Cijk)

and CT = (FjkiG1 jyCjk) is given by (3.5.1.4), where Dlj-κ = gjrDrik and

⅛A-ijk = Iijk ~ Tjki + 7fcij,

Tijk — 9jrTfk.

Theorem 3.5.1.2. Let a linear connection T = (Γ *∙ fc(τ)) be given in a Finsler

space Fn = (Λ'/, L(xy y)). Ifthe associated Finsler connection *Γ = (Γ * λ,, Γθj, Cjk) is L-metrical. then Fn is a generalized Bcrwald space with the generalized Cartan connection *Γ. Proof: We deal with ,Γ0 = (ΓJfc,Γ⅛j∙,O). (2.4.3.1) and (2.4.3.3) gives P1 = 0

and P2 = 0 of *Γ q. Hence (2.2.5.8, b) shows Vh and Vυ = ∂ with respect to *Γq are commutative, and so we observe

v⅛.v>{⅛(≤)}-4⅞{v'∙(⅞)} = o. Hence *Γ is h-metrical, and Γ * is a generalized Cartan connection. From ΓJfc = Tjk(x) it follows that Fn is a generalized Berwald space with *Γ.

3.5.2

Wagner Space

We propose now an interesting class of generalized Cartan connections by taking special skew-symmetric tensor Tjk as follows: Definition 3.5.2.1. Let a covariant vector field Si(x) be given in the underlying

manifold M of a Finsler space Fn = (M1 L(x1 yf). Put

Tjk = φ⅛M - δiksj(x),

and construct a generalized Cartan Connection CT(T) with respect to this T. CT(T) is called a Wagner connection ⅛T(s) with respect to Sj(x). If Fn is a generalized Berwald space with respect to T1 then Fn is called a Wagner space with respect to Si(x). Ref V. Wagner [167]. The name “Wagner space” was given by M. Hashiguchi [46].

Let Tjk (x) be a skew-symmetric tensor field in a two-dimensional manifold. Tf we put T1 — Tfl and observe q^∣ ηrr rπ2 τ∣2 21 — lrl — I2I — -112’

n-t rτ-∣r ml 12 — J-r2 ~ i12>

then Tjk is written as Tjk = δjTk — δkTj. Therefore we have Proposition 3.5.2.1. Any generalized Berwald space of dimension two with

respect to Tjk(x) is a Wagner space with respect to Tfi.

Matsumoto

738

Substituting this TJk = δljsk — δ1 ksj in (3.5.1.4), we get the difference of WΓ(s) from CT : sh=ghrsr,

(a)

¾ = V‰Λ

(b)

Vikh = Aw{gikδ>h + C{kyh} + Ckhtf - C>khyi + L2(¾h + ⅛¾).

(3.5.2.1)

Λs Theorem 3.5.1.1 shows that a Finsler space with HT(s) is a Wagner space, if and only if WT(s) satisfies Cfajik = 0 with respect to WΓ(

(b) Λ⅛ = Λ⅞ + ≡⅛⅜.

Substituting from (3.6.3.1) and (3.6.2.1,b), Nij is written as 2Z∕2∖ oJX ∕

(

This is symmetric and Mo = θ∙ Then, multiplying by ghj, (3.6.3.2) easily yields

Nh ■ Nij = (^Z¾ ’

N=

(3.6.3.3)

½≤1±⅛), (n + l)

(3.6.3.4)

On the other hand, (3.6.3.1) gives

3

while (3.4.4.3) is rewritten in the form

LChij∖k + IhCijk + IiChjk + IjChtk + IkChij = 1 (π2

1)1 U^fii^jk 4“ hhjhki + hhkhij).

Multiplying by ghz, we get LCj∖k + tjCk + lkCj =

,

Matsmnot o

746

which can be rewritten as LCJk

= { (7+T) + (⅛j }hJk +{½ —

χA∕ry _ χAμι∕ 0 ~ 0Γ23 '

which define the so-called ε-tensors : p.

— X j Szjk0θ'3'yCl Sj ek ,

-c,ijk _ — 0sλμv fiiλ)cjμ)ekv)∙

Hence δaβy and δλμ,3 are scalar components of the covariant and contravariant ε-tensors, respectively. Put E = dot(⅛}) and g = det(y%. Consequently, Ha3y + Hjay = 0. Furthermore, e^lj∙ = t1ij = 0, which implies = 0. Thus the matrices Hy = (Haβy) may be written as /0 0 0∖ Hy = (Ha3y) = P 0 hy ∖0 — hy 0 ∕

.

(3.7.1.7)

756 where

Matsumoto

= H23~f∙ Then, mυ = e2)√

= ≤f,3)ej, =

= H23-,930eiβ∙te]i = h,,ε3nicp.

Similarly, rizj = -h^ε2∏ιzε^. Hence, if we introduce the vector field

hj=hye]∖

(3.7.1.8)

awe obtain

riltj = -ε2mzhj.

mzlj = ε3rilhji

(3.7.1.9)

The vector field hj is called the h-connection vector. Next we consider the !/-covariant derivatives e^∣j∙. Denote them as

(3.7.1.10)

i⅛)b = K⅛√∙

In the following most of scalar components are restricted to (0) p-homogeneous. Since eza^ and ⅛Jj are (0) and (-1) p-homogeneous respectively, we shall mul­ tiply the latter by L to get the scalar components, as (3.7.1.10), which are (0) p-homogeneous. Similar to the case of His, if we put Vrαj7 = V⅛ygpβi then gij∖k = 0 leads to the skew-symmetry Vβay = — Vqj7. Next, observe -^el)i∣J =

Ij =

9ij ~

= ^lpσei^∙

Multiplying by ¾e7p the above gives

Viβj5∙-y

C(λβ^gδ∖ H- Cctβδg^ι = 0∙

In the case (7, δ) = (1,2) and (1,3), the above are trivial, because of (1) and (2) OfProposition 3.7.1.2. The case (7, J) = (2,3) yields

C* q/92;3

- Cfα33ζ2 = θ∙

This is trivial for α = 1 from (2) of Proposition 3.7.1.2. Consequently, we put (ot,β) = (2,2), (2,3) and (3,3) : For instance, C,222J3 — C223>2

= (βiC222)^3y —

3Cp22V23 -(⅞C223)⅞ + 2Cp23V∕2 + C22pV3p2

= ε2H∖3 — ⅛C322V233ε3 ÷ ε2J∖2 +2C323V232ε3 —

C⅛22½32-2∙

Similarly,

ε2(J2 ÷ H3) = (H ~ 2εl)v2 — 3εJv3,

ε2(I.2 ÷ J;3) = 3Ju2 + (H- 2εl)υ3i - ε3 J∙2 + ε2L3 =

3(Iv2 + Jv3).

(3.7.3.3)

Finsler Geometry in the 20th-Century

763

Therefore the scalar components Tαp∙7d* of LT}ljjk are T∖β^ = θ anfI T2222 = C2⅛2 ÷ 3εJv2∙f T2223 =

C2H,3 ÷ 3εJvα

= — ε2Jf2 ÷ (H ~ 2εZ)v2? 7⅞233 = — ε2J,3

+ (H — 2εl)υ^

= ^2Λ2 - 3Jv2,

(3.7.3.4)

3 JV3 = ε3J∖2 ÷ 31^2,

T,2333 = β2-f∙,3 “

T3333 —

ε3Jt3

+ 3iv3.

We now consider a three-dimensional Finsler space with the vanishing T-tensor. Compare Theorem 3.7.3.1 with Proposition 3.1.3.1. Theorem 3.7.3.1. Let F3 be a Finsler space of the dimension three with the

non-vanishing Ci. The T-tensor of F3 vanishes, if and only if the v-connection vector v vanishes and all the main scalars are functions of position alone. Proof: T2233 = T2333 = T3333 = 0 imply (■f;2) Λs> ,¼>

*λβ) = (3ε2 Jt,2> 3β2 J⅛, —3ε3⅛2,3ε3Zr3),

and then T2223 = T2233 = 0 yield (J7 + ε∕>2 = (H + εZ>3 = 0.

Since H ÷ εl = 0 contradicts C ≠ 0 from the Proposition 3.7.3.1, we have υ2 = V3 = 0, that is, Vi = 0 from Lemma 3.7.1.1. Then (3.7.3.3) implies ∕f2 = H.3 = I;2 = I,3 = J,2 = J,3 = 0∙ H-1 = /;i = J.1 = 0 from homogeneity. Hence H, I and J do not depend on y.

3.7.4

Curvatures

First, we introduce an operator on skew-symmetric tensors for frequent use. Let Tijk, fθr instance, be a tensor of (0,3)-type which is skew-symmetric in (i,j). Using the ε-tensor given in Example 3.7.1.1, if we define a tensor

*⅛ = ∣εhi‰,

(3.7.4.1)

Tijk — Cijh Tk .

(3.7.4.2)

then we get Ty⅛ in the form

764

Matsumoto

This operation on T1 jk is called the shortening of Tijk - Let us denote by Tag1 and *Tβ the scalar components of Tijk and *T%. Then we have *⅞o = I δapσTpaβ,

Taβ. = δa,3p'T'.

Now we deal with the v-curvature tensor S2 = (Sfajk) ∙ Sfajk — ChkrCij ~~ ChjrC↑k∙

(3.7.4.3)

This is skew-symmetric in {hii) and (j,k). Thus, by double shortening,

*Shi = (l∕4)εhjkεitmS,kem,

Sjki,,i = εjkhεtmf Shi.

Letting * S af, be the scalar components of L2{*S hl), we have *Sa *R pσ + C2∕P3σ - C‰ *fi According as a = 1,2,3, the above is equivalent to

(a) * R23 - * R y2 = 0,

23 - * Λ J(

(b) *7? 31 - * Λ 13 = (PI - * R εI)

(c) * β 12 - *7? 21 = * R I(

33 - *ε R

33 - * Λ ε

22} - * R 2J

22),

(3.7.5.6)

23.

Next we consider (3.7.5.2). If we put λαp = *7zαpj ? +* ¾

(pσ⅛3, _

gσ3 *

r2p^

then (3.7.5.2) has scalar components = θ∙

It is obvious that this is equivalent to the simpler equation ∑(37∂){^7+(⅛)"^∙

= (⅜M¾⅝'

and

LCCijk = ε3IX(w{hijCk} + (H- 3εl) ⅛⅛⅛. Thus we have a semi-C-reducibility, because p/4 = ε3∕∕LC and ε%q = (H — 3εΓ)∕LC satisfy p + q = 1 from Proposition 3.7.3.1. Now we consider a semi-(7-reducible and Landsberg space F3 with C ≠ 0. .Owing to Theorem 3.7.4.2 and 3.7.6.1, we have first

h1 = H1 =Z1=J = O.

Theorem 3.7.5.1 shows that S is h-covariant constant, and hence (2) of The­ orem 3.7.4.1 leads to /ξα÷λ‰=0,

a = 1,2,3,

K = H-2εI.

Rom Phijk = CijkiO = 0 of (1) of Theorem 3.3.3.1, (3.7.4.8) yields C27 5,3, which is written as

Hi3

= ε27Ch2,

J2 — ε2Fh3,

1,3 =

3ε2Ih2.

(3.7.6.2) C3^,2 =

(3.7.6.3)

772

Matsumoto

Hence (3.7.6.2) are rewritten as Ih2K = O,

IHi2 + C2(AT)2Zi3 = O.

(3.7.6.4)

Since I = K = O contradict Proposition 3.7.3.1, (3.7.6.3) and (3.7.6.4) show that the spaces under consideration are divided into three classes as follows: (I) I = 0 : h3 = 0,

(II) K = Oih2 = 0, (III) IK ≠ 0,

H3 = ε2Hh2,

H2 = Z,2 - H3 = Z3 = 0,

h2 = 0 : ZH2 = -ε2(K)2h3i

Z2 = ε2Lh3l

H3 = Z3 = 0.

Further we consider the identities (3.7.3.2): In our case, they are written as

ε2H3 = Kv2,

ε2I2 = K υ3,

ε2I3 = 3Zr2.

If F3 belongs to the class (I), then we have ε2H∙3 = Hv2 and v3 = 0. Further S = O from (2) of Theorem 3.7.4.1. If F3 belongs to the class (II), then H3 = Z2 = O1 and hence H2 = Z3 = 0 from K = 0, and consequently H and Z satisfy H a = H.a = Ifa = Ia = 0. Therefore, Theorem 3.7.6.2. A semi-C-reducible and Landsberg space F3 with non-zero

C has J = Hfι = Ifι = hi = 0 and the h-covariant constant v-scalar curvature S. All of such spaces are divided into three classes as follows: (I) I = 0 : h3 = v3 = 0, H3 = ε2Hh2, H3 = ε2Hv2, S = 0,

(II) K (= H - 2εZ) = 0 : h2 = 0, H, I = const., (III) IK ≠ 0, h2 = 0 : ZH2 = -ε2(∏)⅛3, Z2 = ε2Kh3, H3 = Z3 = 0, H3 = ε2Kv2, L2 = ε2Kv3, Z3 = 3ε2Zv2.

If h3 of F3 belonging to the class (II), then ha = 0, α = 1,2,3 and hence Theorem 3.7.4.3 shows that F3 is a Berwald space. The similar fact holds for F3 belonging to the class (III), because v. *e have ha = 0 and Hfθt = Zα = 0, a = 1,2,3. Therefore we obtain Corollary 3.7.6.1. A semi-C-reducible and Landsberg space with non-zero C

belonging to the classes (II) or (III) is reduced to a Bcrwald space, if and only if h3 = 0.

The remainder of the present section is devoted to detailed discussion of spaces belonging to the class (I). They have interesting properties such that the hυ and v-curvature tensors P and S vanish. Our discussions are especially based on the Bianchi identities (3.7.5.6) and (3.7.5.8). On account of Z = J = 0, the former is simply written in the form *∏23 = *∏ 32,

*H 31H *

13 = H *

23,

*H 12 = *H 21.

(3.7.6.5)

Fiiisler Geometry in the 20th-Century

773

From C%y = O except Cf2 = H, the latter is also written

*Rax + 'RaxC2 ij3 = 0,

λ = l,3,

*Λ¾ 2 = 0∙

(3.7.6.6)

To deal with (3.7.6.6), we observe, for instance, *Λ⅛3 =

(∂i* R a3)ei, + tRpiSaVpoj3 + * RapS3Vp3j3.

In the following we shall regard *R a3, for brevity, as nine single scalars and denote then by Raβ. Similarly, t»2 is regarded as a single scalar and denoted by v, because t»i = υ⅛ = 0. (C0eβy have been regarded as a set of the single scalars in Proposition 3.7.3.1.) Then we may write as Ra3∙β and = Ra3t3 + εa(Rι3ha3

+ ⅛3½α,J + ‰‰fl)

+ ε3(Ralh3β + Ra2Vβ).

Consequently, (3.7.6.6) is written (1) ft√?

= Rθl3∖β + Raiε3h3β + Ra2ε3Vβ + Ra3C23 +Ri3εahaβ ∙+ ‰εα‰ + ‰εα‰^ = 0,

(2) q3 = Rσl∙t3 + RalCl3 ~ Rq2^2,3 FRliεahaβ

(3) q.1?

Ra3^3β

+ ½iεα½αp' ÷ ¾l^α^3α∕3 = 0,

= Ra2tβ + Raiε2h2β — Ra3ε2Vβ ÷ ½2^α½α.3 ÷ ¾2^α½α∕3 = θ∙

We calculate (1)2/2-(3)s,j, (3)ι∕j-(2)2^ and (2)33-(I)i^ to eliminate Rctβ'n . Then (3.7.6.5) yields

R23(ε2Hh2β — C2β)

= 0,

(R23H)lβ + R23HCli

=0.

R23ε2Hvtj — ¾2Cf3 = 0,

The first equations are trivial. The second with β = 3 is trivial, while with β = 2 we get T?i2 = ¾3^2^∙

(3.7.6.7)

The third with β = 3 is a consequence of (3.7.6.7) because of Hi3 = ε2Hv from Theorem 3.7.6.2 and R23⅛ = — R↑2 from (l)23, while with β = 2 it is

(R23H').2 = ⅛3(ff)2∙

(3.7.6∙8)

Then, using (3.7.6.7) and (3.7.6.8) and H.3 = ε2ffv, the above (l)ct0, (2)q^

AIatsumoto

774

and (3)oj are reduced to twelve independent differential equations as follows:

⅞3j3 =

⅞3j2 = — ∙R31 ÷ (⅞3 — ε⅞2)ε2^

¾2j2 = — Rll

~ R12H ÷

I?31;2 = ⅛3^2 -

R3iH ~

Rll∙l2 = %R12ε2 ~

(¾2

÷^⅞1)-2j

- -¾2,

-^12;3 — -¾3≡3,

-R12ε3υ,

‰j3 =

-Rll + R33 -3,

-ftιij3 = 2(¾ι -

¾1⅛

R23H)ε3.

R22t2 = R22,3 = O, ⅞3j2 = -∙2⅞3S3V —

R33,3 =

R33H)

— 2⅞1 + ⅞3^∙

(3.7.6.9) Here we shall recall (2) of Theorem 3.7.4.1. In our case, it gives V2,3-V3f2 = εWe observe V2-,3 = (∂tυ2)ei3j - υpVg3 = υ-3 t,3j2

= (∂iv3)e^ - vpV3p2 = (t')2ε2.

Thus, v;3 = ε + (u)2ε2∙

(3.7.6.10)

Now we consider the integrability conditions Raβ^∙tδ = Raβ-,δ,y of (3.7.6.9). Making use of = ε2Hv and (3.7.6.10), the conditions are written as (υ)2C‰

-εR22) = 0,

υ(R11 — ε2R22 ÷ R12Rr ~ R3ιε2υ) = 0,

v{R23 ~ R31ε2H ~ R12εv) = 0,

(3.7.6.11)

v (2⅛2 — Rnε2H) = 0, v(2R23υ + R33ε3H) — 0.

Since v does not vanish from (3.7.6.10), we have six linear equations (3.7.6.7) and (3.7.6.11) for Raβ. It is easy to show that these equations yield 3 + (II)2c2 + (v)2ε3 = 0 or R33 = 0. If the former holds, then we get HH.3ε2 + vu3ε3 - v{ε2 + (H)2 + (υ)2ε} = -2ε2v = 0, which contradicts (3.7.6.10). Therefore, ⅛3 = 0 and all Raβ = 0.

Finsler Geometry in the 20th-Century

775

Therefore, Theorem 3.7.6.3. Let F3 be a three-dimensional semi-C-reducible and Lands­ berg space with non-zero C. If F3 belongs to the class (I) of Theorem 3.7.6.2,

then the three curvature tensors Ri P and S of the Cartan connection vanish identically. Remark: The main subject of the present section was to generalize The­

orem 3.4.4.1. It is observed that the spaces belonging to the classes (H) and (III) are near to the Berwald spaces. However the spaces belonging to the class (I) gave rise to the troublesome problem: iiR = 0?” Let us recall that R ≠ 0 appears in Theorems 3.6.3.1 and 3.6.4.4.

3.8

Indicatrix and Homogeneous Extension

3.8.1 Indicatrix as Riemannian Hypersurface We consider a Finsler space Fn = (MiL(xiy)). The fundamental function L(i,,2∕) gives the fundamental tensor gij(xiy) = ∂.∙¾∙(L2∕2), which satisfies gij(x,y)ytyj = L2(x.y). As it has been shown in §1.2.2, the tangent vector space Mx at a point x of M is regarded as an n-dimensional Riemannian space equipped with the Riemannian metric gij(x,y)ytyj with fixed x. It follows that the Christoffel symbols of Mx : ∣ 9irΦj9rk + ∂kgrj - ∂r9jk) ■

are nothing but the components CJfc of the C-tensor. Hence, the covariant dif­ ferentiation in this Riemannian space Mx coincides with that of the vertical connection given by Proposition 2.4.1.1, for instance, the Cartan connection CΓ and the Hashiguchi connection HΓ. Further (2.4.3.4) and (2.5.1.2) show that the v-curvature tensor S2 of CT is just the curvature tensor of the Riemannian space Mx. We shall recall the indicatrix Ix = {y ∈ Mx∖L(xiy) = 1} defined in §1.2.2. It is a Iiypersurface of Mxi given by the equation L(xiy) = 1. Let yz = yz(ua)i

a = li...in-1,

be parametric equations of Ix in M : i(j,∙,y(u)) = 1.

(3.8.1.1)

In the following, we restrict our discussions to a Finsler space Fn of dimen­ sion n more than two, and consider Ix as a Iiypersurface of the Riemannian space Mx. From (3.8.1.1) we have first 4¾ = 0,

=

(3.8.1.2)

776

Matsumoto

Bα with the components B1 ct are regarded as n — 1 linearly independent vectors tangent to Ix. The linear independence is, of course, assumed, that is, the matrix (j¾) is of rank n — 1. Equation (3.8.1.2) shows that the normalized supporting element ⅛ = ∂iL is the unit normal of Ix. Since we get the frame field (Bla,Γ) of Mx at every point of Ix, we obtain the dual cofranιe field (Bzq,Λ) : ⅜S∕ = ⅛

(iB? = 0,

BUi = 0, l'ti = 1.

(3.8.1.3)

Also,

BiaB * + titj = δ*.

(3.8.1.4)

On Ix we get the induced Riemannian metric gaβ(u) = gij(τ ,y(u))BtaB3 f3. From (3.8.1.2) this is also 9aβ(u) = hij (z, j∕(u))BiB⅛.

(3.8.1.5)

According to the theory of subspaces in a Riemannian space, we obtain the Gauss and Weingarten derivation equations

(a)

∂βBia + CJ1 (x, 2z(u))Bj Bkp = γj,b; + Haβei, (3.8.1.6)

(b)

∂βC + CJfc (x, y(u))0Bk0 = -BjJBj1

where ∂β = ∂∕∂uP> Ps are Christoffel symbols constructed from g0t3(u), Haβ is the second fundamental tensor of Ix and Hβ = Hβσgσ^r. H is, of course, a symmetric tensor. The second term of the Weingarten equation (b) vanishes because Cjky3: = 0. Next, differentiating (3.8.1.2) by uti, we have

hijB^+Ii∂βB^ = 0.

Paying attention to (3.8.1.5) and substituting from the Gauss equation (a), we get Baβ = -gaβ. Thus, Theorem 3.8.1.1.

The Gauss and Weingarten derivation equations of the

indicatrix Ix are (1)

⅛Bi + Ci kB3 aB0 = rχ3βjt - ga,(∖

(2)

∂βtl = Bj.

Corollary 3.8.1.1. The indicatrix Ix of a Finsler space F", n ≥ 3, is a totally

umbilical hypersurface with the mean curvature —1. Thus Ix may be regarded as the unit sphere in a Euclidean space. X'

-'

-SA >X

Finsler Geometry in the 20th-Century

777

Next we shall find the Gauss and Codazzi equations which are integrability conditions of the Gauss and Weingarten equations. They are ShιjkBaB^B^Bδ

Sh with the total curvature κ. This may be written as Rtxβyδ ~ {θay9βδ

9ocδ9βy} = (κ ~ ^-){9ay9βδ ~ 9ctδ9βy)∙>

and apply the homogeneous extension, then we obtain (1) of Theorem 3.7.4.1 without the assumption C ≠ 0, where S = κ - 1. Since S = O implies κ = 1, we get

779

Finsler Geometry in the 20th-Century

The v-curvature tensor S of the Cartan connection CT vanishes at a point x, if and only if the indicatrix Ix is of constant curvature 1.

Theorem 3.8.2.1.

Roughly speaking, 5 vanishes, if and only if Ix is a unit sphere. We shall give several examples. Theorem 3.8.2.2. The v-curvature tensor S of the Cartan connection of a four-dimensional Finslcr space is written in the form

Shijk = A[jk]{h,hjMik 4^ hikMhj} > where Mij = Sij — Shi3 ∕⅛. Proof: We put

_

Mhijk =

{b>hjMik + h{kMhj}∙

Both Shijk and Mhijk are indicatory and (—2)p-homogeneous. Their projections on Ix are respectively given by

Saβ^l o

and

5∣ {gcn Mβg ÷ gβt

},

where Mβs = Sβg - Sgaδ∕½- It follows from (3.8.1.9) that Saβ = Raβ - 2gcxβ and S = R — 6. Hence, Λ∕α,? = Na0

Na0 = Ra0 - (≤) ga0.

It is well-known that the conformal curvature tensor of a three-dimensional Riemannian space Ix vanishes identically, that is, we have

R∙aβ^δ = ∙^-['yδ]{.9ot'Y^βδ

rojj⅛+^jTtθ⅛+ZfeT⅛jθ)

l

l (titjToQk+fjtkTiOo+tkti'Jojo)

^l^

L2

itij^kT000

L3

Proposition 3.8.3.1. The projection of the indicatorized tensor * T of T on Ix

coincides with that of T on Ix.

Because (3.8.1.2) and (3.8.1.3) lead to hl-Bza = B*f t and h⅛Bfi = Bf, we get {Tijhh ihPk)B^Bkg = TijBJB}s.

Finsler Geometry in the 20th-Century

781

Theorem 3.8.3.1.

(1) The indicatorized tcnsoi' *C∏ j∙∣fc) of the h-coυariant derivative Tiilk in CΓ of an indicatory tensor Tij is given by *(rp ∖ _ Tijtk — TijiQlk {J-Zj>k') — ∙

(2) The indicatorized tensor *(T ij∖k) of the υ-covariant derivative Tij∖k in CT of an (r) p-homogeneous indicatory tensor Tij is given by

+

*(Ty∣t) = Ty∣fc +

φ

Proof: (1) From (3.8.1.3) it follows that

*(Ty.fc)

= Thlmh,>hljhf = (TMh^h‰h% = Tijirnhr∣f = Tijik — lkTijiQ∣L.

(2) We have τij∖mh'kn

= ⅞∣fc-(f)τ√l-,

Tmj∖kh?

= Ty∣fc rp I

((^τnjj∕m)lfc-Tmj J∕7n∣fc)⅞

= Trj∖k--------------------T,------------

_ T, I

I T,kjtj

-Ti3∖lc+ -£-•

Consequently,

Trs∖thrihsjhtk

= ITrslfc - (f )τrA}Khsj = (τis!fc + ¾ii)⅛j - (f )Tj√,fc = ⅞lfc + ⅞^ + 2≠-(f)r√fc∙

Proposition 3.8.3.2.

(1) The indicatorized tensor *(gij) of the fundamental tensor gij∖ is the angular metric tensor hij. (2) The indicatorized tensor *{LChij∖k) of LChij∖k is the T- tensor Tflljk-

Matsunioto

782

Proof: (1) gtjhlhtijc = hhk is obvious. (2) It is a consequence of (2) of The­ orem 3.8.3.1.

Finally v.τe consider the vanishing of the T-tensor. We denote by Λα∙>π the covariant derivative of a tensor field Kaβ on the indicatrix Iτ with respect to the induced connection (Γ'^7) in (1) of Proposition 3.8.1.1. For the projection Kaβ = K0B * + Kij(-σrsBraB° + Γ * 7Bj - gα√ *)B +M(-(¾∣ΛA + rsθyBi - gβJi)

jj

= (Kij.k - KrjCrk - Ktrqk)Bi,B^ +Kii)Γδ aι + κasrδh - k°=s∙--'b∙∙ -

,

which shows

κaβ, 1 = Kij∖kB⅛⅛B * -

+

(3 8 3 υ

Now, applying this formula to the C-tensor, we get Ca0τ,f = Chij∖kB^Bif,B^Bξ.

On account of (2) of Proposition 3.8.3.2 and L = 1, we obtain

ThijkB⅛⅛Bi,B%

= (Cpgr∖sh^h^BiβBilB^ = Cpqr∖sBlB^Bl,

and hence we have Caβr,t = Th,,kB^BiβBifB%.

(3.8.3.2)

Since the T-tensor is indicatory, Proposition 3.8.2.1 shows Theorem 3.8.3.2.

(1) The T-tensor vanishes, if and only if the projection of the C-tensor is covariant constant on Ix. (2) If the T-tensor vanishes, then Ix is locally symmetric in the sense of Riemannian geometry. Proof: (2) Caβrj - 0 implies Rυtβ^^s — 0 from (3.8.1.7) and (3.8.1.8). Theorem 3.8,3.3. If the T-tensor of a three-dimensional Finslcr space with

non-zero C vanishes, then Ix is locally flat. Proof: Theorem 3.7.3.1 implies t√ = 0 and (2) of Theorem 3.7.4.1 gives S =

-1. Hence (1) of Theorem 3.7.4.1 yields Saβ^. = -(ga^fgβι - gaδg>h) by the projection on Ix. Thus (3.8.1.7) yields = 0.

Chapter 4

Conformal and Projective Change 4.1 4.1.1

Conformal Change Geometrical Meaning of Conformal Change

In §1.2.2 the absolute length ∣v∣ of a tangent vector v at a point x of a Finsler space Fn = (M,L(xiyf) is defined as the value L(xiv)f provided that L(x>y) > O for any y. On the other hand, if g⅛ (a?, y)ξlζj is positive-definite, then we get the length ∣v∣3z of v relative to y which is equal to yz⅛∙j (a?, y)vivi . As to the angle θ between two vectors u and v at xi COS 0 = -r_____

y/9ij(xtu)υW y∕gij(x,u)vzυJ Consequently, the notion of the angle is not symmetric. Thus, the above θ may be called the angle of v with respect to u. If we take y as uy then we get the angle of v with respect to y as

L(τ,y)y∕gij(x,y)viυi ’ where yi = gi(x,y)yj. Definition 4.1.1.1. Let Fn = (Mi L(x,y)) and * F n = (M, *(x,y)')

be two Finsler spaces on the same underlying manifold M. If the angles of any tangent vector, with respect to any y in Fn and * Fn, are equal to each other then * F n is called conformal to Fn and the change L → *L of the metrics is called a conformal change of metrics. 783

784

Matsuinoto

Hence * F ,'i is conformal to Fn, if and only if 'L2(,x,y) *g

pq(x,y)υpv'l(yrVr)(.ysυs)

(11 1 1)

- L2(x,y)gpq(x, y)vpvq(*y τvr){"ysυs) = 0

holds for any x, y and v. The differentiation by υh, v∖ υi and υk of (4.1.1.1) yield the equation con­ sisting of six terms each of which is of the form 'L2(x,y)"ghi(χ,y)yjyk

- L2(x,y)ghi(χ,y)'y * jyk,

which may be rewritten in the form

*L2(x,y),hhi(x,y)yjyk - L2(x,y)hhi(x,y) *y

jyk.

Thus we obtain

iΛ * ‰){

,⅛⅛!⅛ + 'hjkyhyi) ~ L2(h * hiy* jyk + h * kykyi)} = 0.

If we multiply by yh and next by yk, then the above gives

¾jk){‰Vk - h↑3yk} = 0,

*hij = τ2hij,

where τ = *L∕L. Thus the former is rewritten as ^(ijk){hij(τ2yk - *y k)} = 0.

Multiplication by gli leads to Z⅛∕L = * ⅛∕ L and hence = τ2gij. Putting τ = ec(χ"y∖ we obtain = e2cpjj. We must pay attention to g⅛ = ft⅛(L2∕L), which implies

∂k9ij = Θj9ik

(= 2C *ijfc).

Hence, we get ∂∣cc = 0, immediately. Therefore, we have *L = ec^L. Conversely, this relation gives *g⅛ = e2cgij and i*⅛ = e2c^, and hence (4.1.1.1.) holds. Therefore, Proposition 4.1.1.1. The change L →* L of metrics is conformal, if and only

if we have a function c(x) of position alone, satisfying *L = ec'^L. The function c(x) is called a conformal factor of the conformal change L →

*1. Remark: As above, we got c = c(α?) from vgij = e2cgij. If we first suppose

*L = ecL with c = c(x.y), then this does not impose any restriction to this change L → *L, because ec = *L∕ L only defines c(x,y) for any L and *L.

Finsler Geometry in the 20th-Century

785

From the conformal change L → *L = ec^L, we obtain

(a) *g ij = e2cgij, *g t> = e~2cgi∖

(4.1.1.2)

(b) , j∕i = e2cyi, (c) ∙L, = e~cti. *C l = ecC,,

(d) *h ij = e2'hij. Next we have C ijk=cijk. *

*Cijk = e2cCijk,

(4.1.1.3)

Thus the C-tensor C = (Cjk) is a conformal invariant. In the two-dimensional case, (4.1.1.2) and (3.1.1.2,d) yield

⅛ = ecπii,

*m τ — e~cmz.

(4.1.1.4)

Then (4.1.1.3) and (3.1.1.10) lead to Proposition 4.1.1.2. The main scalar I of a two-dimensional Finsler space is

a conformal Invariaiit. Ref The theory of conformal changes of Finsler spaces has been studied from the early period of Finsler geometry. See M.S. Knebelman [78] and S. Golab [43, 44]. New theories have been developed since M. Hashiguchi [47].

4.1.2 Conformal Changes OfLandsberg and Berwald Spaces We consider a conformal change L → *L = ec^L of metrics. Putting F = L2∕2, we have * F = e2cF and log * F = log F+2c(x). Thus we get a series of conformal invariants: Oi log F, ∂i∂j log F, ∂i∂j∂k log F,... . In particular, Bij = ∂i∂j IogF, is written as

Bij = {9ii

■

(4.1.2.1)

The matrix (¾j ) is non-singular and has the inverse (Blj) given by

Bi> = F(gij - 2t7j).

(4.1.2.2)

On account of (1.2.2.2) we get the important fundamental relation *Gi = Gi - Bir cr,

cr = ∂rc(x).

(4.1.2.3)

Matsunioto

786

Then the relations between quantities of the Berwald connections are as follows:

(a) * G}=G}-

Bif = ∂jB'r,

Bfcr,

(b) 'G≈fc = Gjfc-BjJcri

(4.1.2.4)

Bjrfc = ¾Bjr,

(c) * Gjfcfc = Gjfcfc - Bjfcfccr,

Bjrfcfc = ⅞Bjrfc.

Next, from (4.1.1.3) and (4.1.2.4) we have ‰=¾ Li) is an interesting example of a 3-dimensional Berwald space. er(l)Lj = ec(l){(i∣1)3 + (3∕2)3 ÷ (y3)3 - 3y1y2y3}1,j.

(⅛L2): (L2)4 = 6(2∕1)W∙

(2)

The typical α2j∙ of this quartic metric are L2αl1 = 2y2y3,

L2aj2 = 2yly3,

a22 = O,

L2a23 = (y1)2,

and hence

L2a11 = -(y1)21

L2a12 = 2y1y2,

L2a22 = -4(√2)2,

L2α23 = 2y2y3.

Therefore (R%,L2) is conformally closed as a Berwald space, and is a 3-dimensional Berwald space.

(7¾∖ L3) : (L3)- = n∖yly2 ...yn,

(3)

(Rq i

ec^L2)

n ≥ 3.

This n—th root metric of dimension n is called the Berwald-Moor metric (Pro­ position 3.2.1.2). It is easy to show

L2α11 = -n(n - 2)y1y2, Thus

(Rq,cc^L3)

(4)

L2ai2 = ny1y2.

is an n-dimensional Berwald space for any c(x).

(Rq,

L4) : (L4)rn = (√)"1 + ∙ ∙ ∙ + (yn)rn,

This metric is called the ecological metric. Clearly,

n ≥ 3.

Matsuinoto

790

Thus (Bn,L4) is not conformally closed as a Berwald space. Namely, there exists a function c(x) such that (Bθ,ecZ4) is not a Berwald space. Let us consider two-dimensional Berwald spaces. By Theorem 3.4.2.1 the set B (2) of all Berwald spaces of dimension two is the direct sum B(2) = B1(2) + B2(2) + B3(2), Bι(2) = {F2

∈B(2)∣F = 0,

I ≠ const.},

B2(2) = {F2

∈B(2)∣Λ = 0,

I = const.},

B3(2) = {F2 ∈B(2)∣Λ ≠ 0, I = const.}, Λ∕(2) = Bi (2) + Bi (2) ∙ ∙ ∙ locally Minkowski spaces.

Since the main scalar I is conformally invariant, the set Bc(2) of conformally closed F2 ∈ B (2) contains B2 (2) ÷ B3 (2). Conversely, F2 ∈ Bc (2) has the vanishing T-tensor on account of Corollary 4.1.2.2, and hence Z2 = 0 from (3.1.3.13). Further Iti = Z2 = 0, that is, I = const. Therefore we have Theorem 4.1.3.2. A Berwald space of dimension two is conformally closed, if

and only if the main scalar is constant. That is,

Bc(2) = B2(2) + B3(2), where Bc(2) = {conformally closed Berwald spaces of dimension two}.

Ref The notion of conformal closeness was introduced by M. Matsumoto [117].

4.2 Conformally Flat Finsler Space 4.2.1 Conformally Invariant HMO-Connection We are concerned with (4.1.2.4,c):

,⅛ =~ Gjfcft - B⅛lccr.

(4.2.1.1)

Definition 4.2.1.1. A tensor field Sjkh(x, y) of (3,0)-type is called B-contracting,

if

(1) it is conformally invariant, and (2) βir = Bi⅛hSikh is regular: det (∕3ιr) ≠ O.

The inverse matrix of (βιτ) is denoted by (Φir).

791

Finsler Geometry in the 20th-Century Example 4.2.1.1. From (4.1.1.2) and (4.1.1.3) it follows that

L4Cjkh,

L4gjkCh,

L3gjklh,

are conformally invariant, but L3gjkCh does not satisfy the condition (2) because of BjkhCh = 0 from the homogeneity and Bjkh = ∂hBjrk. To examine the condition (2) about the other two, wτe consider the twodimensional case. From (4.1.2.8) B^=yjgir-δiyr-δryi-L2CV',

B⅛ = girgjk -

∑0∙fc)∙m + 2cr^∙} + l'2c‰

B‰ = 2girCjkh - 2∑(jkhj{gjkcy + yjC%h} -L2qrk.h, where ∑(j∙fc) denotes the interchange of (j, fc) and summation. (3.1.1.10), (3.1.1.9) and (3.1.1.11) we have

Next, from

L2Cjrk = (I∙2 - εl2)mιrmjk - Imtr(Cm)jk - I(Cm)ιrmjk,

where we used the symbols and similar symbols appearing in §2.1.4. Further we put Iij = Wj,

mjkh = mjmkmh,

(Cmm)jkh = Σ>(jhk){Cjmkmh},

and obtain L3Cirk.h = {(L2-t2 ~ 2εII∙2 - ⅛εl)mir - 2L2(Cm)ir + 2ICir}mjkh + 2{Z(∕⅛π)ιr - (L2 - εl2)mtr}(Cmm)jkh

+ 2Imιr (CCm)jkh. Consequently,

(a)

LBVkh = D"mjkh,

(b)

Dir = 2L2(Cm)ir - (L2.2 - 2εII∙2)mir.

(4.2.1.2)

On account of Example 3.1.1.2, we get det (Dtr) = -4(I.2)2. Assume Sjkh has non-zero mjmkmh-components S. Then BjrkhSjkh=εSDtr∕L= βιr, and hence Proposition 4.2.1.1. In the two-dimensional case, a conformally invariant

Sjh is B-contracting, if and only if Sjkh has non-zero mjmkmh-components, provided that L2 ≠ 0.

We shall return to the case of general dimension. Let us assume that Fn has a B-contracting tensor field Sjkh. Then (4.2.1.1) is rewritten as *G}khS^ = GijkhS^h-βircr,

Matsunioto

792 which is solved for c, as

ci = φi-* φ

i,

φi = ΦirGτjkhS^h.

(4.2.1.3)

Here Sjkh, Bjrkh, anf^ Φ⅛ are conformally invariant. Substituting this cl in (4.1.2.4) , we obtain the following conformally invariant quantities cG? = G'/ - S^r,

cG⅛ = G⅛∙ - B%φr,

(4.2.1.4)

c-⅛¾.

The (v) 7ι-torsion tensor ,R1 is given in the form 'Rjk = ^jk + ⅛w{ytδksj - (∂rN])yrsk - sjNk}.

If our discussions are restricted to a p-homogeneous FΓ, then Corollary 2.4.7.1 yields (∂rN'f)yr = and hence

'⅛ = Tljfc + yi(∂ksj - ∂jsk). Similarly, we have (∂rFjk)yr = 0 and obtain the changed h,-curvature tensor 'Tl2 as ‰ = 7¾k + (δth + U⅛)(∂ksj - ∂jsk).

Theorem 2.4.7.2 shows C7t⅞ = 0, provided that FT satisfies the [Tg-condition. We obtain Theorem 4.3.1.1. If a Finsler connection FT of Fn is p-homogeneous and

satisfies the Uz-Conditioni then the connection ,FΓ given from FΓ by one-sides projective change with respect to sι(x) has the (v) h-torsion tensor ,R1 and the h-curvature tensor ,R2 such that

' ⅛k = RiJk + 2∕i',Λ.

*√fc - ¾∙,J -■ 9J Sk >

'R-hjk = R-hjk + ¾sJfe∙ Corollary 4.3.1.1. Let FΓ be a p-homogeneous Finsler connection satisfy­

ing the U2~condition. The Finsler connection ,FΓ given from FΓ by one-sided projective change with respect to a gradient vector field Si(x) has the same (v) h-torsion tensor R1 and the h-curvature tensor R2 with those of FΓ.

Matsumoto

798

4.3.2

Conformal Change of Wagner Space

We consider now a conformal change L -→* L = ec^L of a Finsler space Fn = (A∕, L) with the Cartan connection CT = (Fjk,Gzj, Czjk). We can introduce a one-sided projective change with respect to ci = ¾c(x) : 'F≈t = ⅛ + -covariant derivatives Sti and S.i of S are given by St = ∂iS - (∂rS)Gri,

S.i = ∂iS.

In terms of these covariant derivatives, ρj∙ can be written as 2⅛∙(S) = S0S.j∙ + 2(SS.j.r + S.jS.r)Gr - Skj(S), where the new operator kj is defined by

kj(S) = S.j-yrS.tr.j.

(4.4.1.2)

Now, let * F n — (M * L) be another Finsler space on the same underlying manifold M. Then, gj( *L) =G * j, which is given as

*G 2

i =* L 0*t j + 2( *h jr +

- *L), ⅛∙(

Matsumoto

800

where Gr appears, not * G r y and * L ;o is in Fn, not in ' Fn. Paying attention to *A>r + *tf r = *g jr and *g tj i, = Vl*L, the above may be rewritten in the form *G 2

i = 2Gi +

- 'L g *

iikj(*L).

(4.4.1.3)

Proposition 4.4.1.1. We have the relation (4.4.1.3) between G1 and * G l of

Finsler spaces Fn = (MyL) and * F n = (M L) * on the same underlying mani­ fold My where ’ L-j is the h-covariant derivative of *L in F'i.

We treat of the last term of (4.4.1.3). Let us introduce one more operator kij by ’ ⅛(S) [= ∂j{ki(S)}] = Svi.j - S.j.z - yrS∙rij . (4.4.1.4) Since (2.5.5.6,b) yields S->.j = S.j-i and

yrS..r.i.j = yrS.i.,r.j = yτ(S.i.j.r - S.kGlj) = yrS.i.j.,r, we have

kij(S) = S.j.i - S.i:i - yrS.i.j.r,

4.4.2

(4.4.1.5)

Metrics in Projective Relation

We deal with the differential equations of geodesic (1.2.2.1). For any parameter t preserving the orientation of the geodesic, that is, ds/dt > 0, they may be written as d2xl + 2G ( ,^dx ~dP * ’ dx di)∖ ^ ^>W ~dtz ’