An introduction to Riemann-Finsler geometry / D. Bao, S.-S. Chern, Z. Shen.

Format:

Book

Author/Creator:

Bao, David Dai-Wai.

Contributor:

Chern, Shiing-Shen, 1911-2004.

Shen, Zhongmin, 1963-

Series:

Graduate texts in mathematics ; 200.

Graduate texts in mathematics ; 200

Language:

English

Subjects (All):

Geometry, Riemannian.

Finsler spaces.

Physical Description:

xx, 431 pages : illustrations ; 25 cm.

Place of Publication:

New York : Springer, [2000]

Summary:

In Riemannian geometry, measurements are made with both yardsticks and protractors. These tools are represented by a family of inner-products. In Riemann-Finsler geometry (or Finsler geometry for short), one is in principle equipped with only a family of Minkowski norms. So ardsticks are assigned but protractors are not. With such a limited tool kit, it is natural to wonder just how much geometry one can uncover and describe? It now appears that there is a reasonable answer. Finsler geometry encompasses a solid repertoire of rigidity and comparison theorems, most of them founded upon a fruitful analogue of the sectional curvature. There is also a bewildering array of explicit examples, illustrating many phenomena which admit only Finslerian interpretations. This book focuses on the elementary but essential items among these results. Much thought has gone into making the account a teachable one.

Contents:

Part 1 Finsler Manifolds and Their Curvature 1

Chapter 1 Finsler Manifolds and the Fundamentals of Minkowski Norms 1

1.0 Physical Motivations 1

1.1 Finsler Structures: Definitions and Conventions 2

1.2 Two Basic Properties of Minkowski Norms 5

1.2 A. Euler's Theorem 5

1.2 B. A Fundamental Inequality 6

1.2 C. Interpretations of the Fundamental Inequality 9

1.3 Explicit Examples of Finsler Manifolds 14

1.3 A. Minkowski and Locally Minkowski Spaces 14

1.3 B. Riemannian Manifolds 15

1.3 C. Randers Spaces 17

1.3 D. Berwald Spaces 18

1.3 E. Finsler Spaces of Constant Flag Curvature 20

1.4 The Fundamental Tensor and the Cartan Tensor 22

Chapter 2 The Chern Connection 27

2.1 The Vector Bundle [pi]*TM and Related Objects 28

2.2 Coordinate Bases Versus Special Orthonormal Bases 31

2.3 The Nonlinear Connection on the Manifold TM \ 0 33

2.4 The Chern Connection on [pi]*TM 37

2.5 Index Gymnastics 44

2.5 A. The Slash (...)[subscript

2.5 B. Covariant Derivatives of the Fundamental Tensor g 45

2.5 C. Covariant Derivatives of the Distinguished l 46

Chapter 3 Curvature and Schur's Lemma 49

3.1 Conventions and the hh-, hv-, vv-curvatures 49

3.2 First Bianchi Identities from Torsion Freeness 50

3.3 Formulas for R and P in Natural Coordinates 52

3.4 First Bianchi Identities from "Almost" g-compatibility 54

3.4 A. Consequences from the dx[superscript k] [logical and] dx[superscript l] Terms 55

3.4 B. Consequences from the dx[superscript k] [logical and] 1/F[delta]y[superscript 1] Terms 55

3.4 C. Consequences from the 1/F[delta]y[superscript k] [logical and] 1/F[delta]y[superscript 1] Terms 56

3.5 Second Bianchi Identities 58

3.6 Interchange Formulas or Ricci Identities 61

3.7 Lie Brackets among the [delta]/[delta]x and the F[characters not reproducible] 62

3.8 Derivatives of the Geodesic Spray Coefficients G[superscript i] 65

3.9 The Flag Curvature 67

3.9 A. Its Definition and Its Predecessor 68

3.9 B. An Interesting Family of Examples of Numata Type 70

3.10 Schur's Lemma 75

Chapter 4 Finsler Surfaces and a Generalized Gauss-Bonnet Theorem 81

4.1 Minkowski Planes and a Useful Basis 82

4.1 A. Rund's Differential Equation and Its Consequence 83

4.1 B. A Criterion for Checking Strong Convexity 86

4.2 The Equivalence Problem for Minkowski Planes 90

4.3 The Berwald Frame and Our Geometrical Setup on SM 92

4.4 The Chern Connection and the Invariants I, J, K 95

4.5 The Riemannian Arc Length of the Indicatrix 101

4.6 A Gauss-Bonnet Theorem for Landsberg Surfaces 105

Part 2 Calculus of Variations and Comparison Theorems 111

Chapter 5 Variations of Arc Length, Jacobi Fields, the Effect of Curvature 111

5.1 The First Variation of Arc Length 111

5.2 The Second Variation of Arc Length 119

5.3 Geodesics and the Exponential Map 125

5.4 Jacobi Fields 129

5.5 How the Flag Curvature's Sign Influences Geodesic Rays 135

Chapter 6 The Gauss Lemma and the Hopf-Rinow Theorem 139

6.1 A. The Gauss Lemma Proper 140

6.1 B. An Alternative Form of the Lemma 142

6.1 C. Is the Exponential Map Ever a Local Isometry? 143

6.2 Finsler Manifolds and Metric Spaces 145

6.2 A. A Useful Technical Lemma 146

6.2 B. Forward Metric Balls and Metric Spheres 148

6.2 C. The Manifold Topology Versus the Metric Topology 149

6.2 D. Forward Cauchy Sequences, Forward Completeness 151

6.3 Short Geodesics Are Minimizing 155

6.4 The Smoothness of Distance Functions 161

6.4 A. On Minkowski Spaces 161

6.4 B. On Finsler Manifolds 162

6.5 Long Minimizing Geodesics 164

6.6 The Hopf-Rinow Theorem 168

Chapter 7 The Index Form and the Bonnet-Myers Theorem 173

7.1 Conjugate Points 173

7.2 The Index Form 176

7.3 What Happens in the Absence of Conjugate Points? 179

7.3 A. Geodesics Are Shortest Among "Nearby" Curves 179

7.3 B. A Basic Index Lemma 182

7.4 What Happens If Conjugate Points Are Present? 184

7.5 The Cut Point Versus the First Conjugate Point 186

7.6 Ricci Curvatures 190

7.6 A. The Ricci Scalar Ric and the Ricci Tensor Ric[subscript ij] 191

7.6 B. The Interplay between Ric and Ric[subscript ij] 192

7.7 The Bonnet-Myers Theorem 194

Chapter 8 The Cut and Conjugate Loci, and Synge's Theorem 199

8.2 The Cut Point and the First Conjugate Point 201

8.3 Some Consequences of the Inverse Function Theorem 204

8.4 The Manner in Which c[subscript y] and i[subscript y] Depend on y 206

8.5 Generic Properties of the Cut Locus Cut[subscript x] 208

8.6 Additional Properties of Cut[subscript x] When M Is Compact 211

8.7 Shortest Geodesics within Homotopy Classes 213

8.8 Synge's Theorem 221

Chapter 9 The Cartan-Hadamard Theorem and Rauch's First Theorem 225

9.1 Estimating the Growth of Jacobi Fields 225

9.2 When Do Local Diffeomorphisms Become Covering Maps? 231

9.3 Some Consequences of the Covering Homotopy Theorem 235

9.4 The Cartan-Hadamard Theorem 238

9.5 Prelude to Rauch's Theorem 240

9.5 A. Transplanting Vector Fields 240

9.5 B. A Second Basic Property of the Index Form 241

9.5 C. Flag Curvature Versus Conjugate Points 243

9.6 Rauch's First Comparison Theorem 244

9.7 Jacobi Fields on Space Forms 251

9.8 Applications of Rauch's Theorem 253

Part 3 Special Finsler Spaces over the Reals 257

Chapter 10 Berwald Spaces and Szabo's Theorem for Berwald Surfaces 257

10.1 Berwald Spaces 258

10.2 Various Characterizations of Berwald Spaces 263

10.3 Examples of Berwald Spaces 266

10.4 A Fact about Flat Linear Connections 272

10.5 Characterizing Locally Minkowski Spaces by Curvature 275

10.6 Szabo's Rigidity Theorem for Berwald Surfaces 276

10.6 A. The Theorem and Its Proof 276

10.6 B. Distinguishing between y-local and y-global 279

Chapter 11 Randers Spaces and an Elegant Theorem 281

11.0 The Importance of Randers Spaces 281

11.1 Randers Spaces, Positivity, and Strong Convexity 283

11.2 A Matrix Result and Its Consequences 287

11.3 The Geodesic Spray Coefficients of a Randers Metric 293

11.4 The Nonlinear Connection for Randers Spaces 298

11.5 A Useful and Elegant Theorem 301

11.6 The Construction of y-global Berwald Spaces 304

11.6 A. The Algorithm 304

11.6 B. An Explicit Example in Three Dimensions 306

Chapter 12 Constant Flag Curvature Spaces and Akbar-Zadeh's Theorem 311

12.1 Characterizations of Constant Flag Curvature 312

12.2 Useful Interpretations of E and E 314

12.3 Growth Rates of Solutions of E + [lambda]E = 0 320

12.4 Akbar-Zadeh's Rigidity Theorem 325

12.5 Formulas for Machine Computations of K 329

12.5 A. The Geodesic Spray Coefficients 329

12.5 B. The Predecessor of the Flag Curvature 330

12.5 C. Maple Codes for the Gaussian Curvature 331

12.6 A Poincare Disc That Is Only Forward Complete 333

12.6 A. The Example and Its Yasuda-Shimada Pedigree 334

12.6 B. The Finsler Function and Its Gaussian Curvature 335

12.6 C. Geodesics; Forward and Backward Metric Discs 336

12.6 D. Consistency with Akbar-Zadeh's Rigidity Theorem 341

12.7 Non-Riemannian Projectively Flat S[superscript 2] with K = 1 343

12.7 A. Bryant's 2-parameter Family of Finsler Structures 343

12.7 B. A Specific Finsler Metric from That Family 345

Chapter 13 Riemannian Manifolds and Two of Hopf's Theorems 351

13.1 The Levi-Civita (Christoffel) Connection 351

13.2 Curvature 354

13.2 A. Symmetries, Bianchi Identities, the Ricci Identity 354

13.2 B. Sectional Curvature 355

13.2 C. Ricci Curvature and Einstein Metrics 357

13.3 Warped Products and Riemannian Space Forms 361

13.3 A. One Special Class of Warped Products 361

13.3 B. Spheres and Spaces of Constant Curvature 364

13.3 C. Standard Models of Riemannian Space Forms 366

13.4 Hopf's Classification of Riemannian Space Forms 369

13.5 The Divergence Lemma and Hopf's Theorem 376

13.6 The Weitzenbock Formula and the

Bochner Technique 378

Chapter 14 Minkowski Spaces, the Theorems of Deicke and Brickell 383

14.1 Generalities and Examples 383

14.2 The Riemannian Curvature of Each Minkowski Space 387

14.3 The Riemannian Laplacian in Spherical Coordinates 390

14.4 Deicke's Theorem 393

14.5 The Extrinsic Curvature of the Level Spheres of F 397

14.6 The Gauss Equations 399

14.7 The Blaschke-Santalo Inequality 403

14.8 The Legendre Transformation 406

14.9 A Mixed-Volume Inequality, and Brickell's Theorem 412.

Notes:

Includes bibliographical references (pages [419]-425) and index.

ISBN:

038798948X

OCLC:

42476109