Introduction To Modern Finsler Geometry

by Yi-bing Shen, Zhongmin Shen

Overview

This comprehensive book is an introduction to the basics of Finsler geometry with recent developments in its area. It includes local geometry as well as global geometry of Finsler manifolds.In Part I, the authors discuss differential manifolds, Finsler metrics, the Chern connection, Riemannian and non-Riemannian quantities. Part II is written for readers who would like to further their studies in Finsler geometry. It covers projective transformations, comparison theorems, fundamental group, minimal immersions, harmonic maps, Einstein metrics, conformal transformations, amongst other related topics. The authors made great efforts to ensure that the contents are accessible to senior undergraduate students, graduate students, mathematicians and scientists.

Product Details

ISBN-13:	9789814713160
Publisher:	World Scientific / Higher Education Press, China
Publication date:	04/16/2016
Pages:	408
Product dimensions:	5.90(w) x 8.90(h) x 1.00(d)

Table of Contents

Preface v

Foundations 1

1 Differentiable Manifolds 3

1.1 Differentiable manifolds 3

1.1.1 Differentiable manifolds 3

1.1.2 Examples of differentiable manifolds 3

1.2 Vector fields and tensor fields 6

1.2.1 Vector bundles 6

1.2.2 Tensor fields 7

1.3 Exterior forms and exterior differentials 9

1.3.1 Exterior differential operators 9

1.3.2 De Rham theorem 10

1.4 Vector bundles and connections 10

1.4.1 Connection of the vector bundle 10

1.4.2 Curvature of a connection 12

Exercises 13

2 Finsler metrics 15

2.1 Finsler metrics 15

2.1.1 Finsler metrics 15

2.1.2 Examples of Finsler metrics 17

2.2 Cartan torsion 22

2.2.1 Cartan torsion 22

2.2.2 Deicke theorem 24

2.3 Hilbert form and sprays 28

2.3.1 Hilbert form 28

2.3.2 Sprays 30

2.4 Geodesies 31

2.4.1 Geodesies 31

2.4.2 Geodesic coefficients 32

2.4.3 Geodesic completeness 33

Exercises 36

3 Connections and Curvatures 39

3.1 Connections 39

3.1.1 Chern connection 39

3.1.2 Berwald metrics and Landsberg metrics 42

3.2 Curvatures 44

3.2.1 Curvature form of the Chern connection 44

3.2.2 Flag curvature and Ricci curvature 45

3.3 Bianchi identities 47

3.3.1 Covariant differentiation 47

3.3.2 Bianchi identities 51

3.3.3 Other formulas 53

3.4 Legendre transformation 54

3.4.1 The dual norm in the dual space 54

3.4.2 Legendre transformation 55

3.4.3 Example 56

Exercises 57

4 S-Curvature 59

4.1 Volume measures 59

4.1.1 Busemann-Hausdorff volume element 59

4.1.2 The volume element induced from SM 62

4.2 S-curvature 65

4.2.1 Distortion 65

4.2.2 S-curvature and E-curvature 65

4.3 Isotropic S-curvature 68

4.3.1 Isotropic S-curvature and isotropic E-curvature 68

4.3.2 Randers metrics of isotropic S-curvature 69

4.3.3 Geodesic flow 72

Exercises 75

5 Riemann Curvature 77

5.1 The second variation of are length 77

5.1.1 The second variation of length 77

5.1.2 Elements of curvature and topology 80

5.2 Scalar flag curvature 83

5.2.1 Schur theorem 83

5.2.2 Constant flag curvature 85

5.3 Global rigidity results 89

5.3.1 Flag curvature with special conditions 89

5.3.2 Manifolds with non-positive flag curvature 91

5.4 Navigation 94

5.4.1 Navigation problem 94

5.4.2 Randers metrics and navigation 99

5.4.3 Ricci curvature and Einstein metrics 104

Exercises 112

Further Studies 115

6 Projective Changes 117

6.1 The projective equivalence 117

6.1.1 Projective equivalence 117

6.1.2 Projective invariants 119

6.2 Projectively flat metrics 123

6.2.1 Projectively flat metrics 123

6.2.2 Projectively flat metrics with constant flag curvature l26

6.3 Projectively flat metrics with almost isotropic S-curvature 136

6.3.1 Randers metrics with almost isotropic S-curvature 136

6.3.2 Projectively flat metrics with almost isotropic S-curvature 141

6.4 Some special projectively equivalent Finsler metrics 144

6.4.1 Projectively equivalent Randers metrics 144

6.4.2 The projective equivalence of (α, β)-metrics 148

6.4.3 The projective equivalence of quadratic (α, β)metrics 152

Exercises 155

7 Comparison Theorems 157

7.1 Volume comparison theorems for Finsler manifolds 157

7.1.1 The Jacobian of the exponential map 157

7.1.2 Distance function and comparison theorems 162

7.1.3 Volume comparison theorems 165

7.2 Berger-Kazdan comparison theorems 174

7.2.1 The Kazdan inequality 174

7.2.2 The rigidity of reversible Finsler manifolds 180

7.2.3 The Berger-Kazdan comparison theorem 185

Exercises 190

8 Fundamental Groups of Finsler Manifolds 191

8.1 Fundamental groups of Finsler manifolds 191

8.1.1 Fundamental groups and covering spaces 191

8.1.2 Algebraic norms and geometric norms 194

8.1.3 Growth of fundamental groups 199

8.2 Entropy and finiteness of fundamental group 205

8.2.1 Entropy of fundamental group 205

8.2.2 The first Betti number 209

8.2.3 Finiteness of fundamental group 213

8.3 Gromov pre-compactness theorems 218

8.3.1 General metric spaces 218

8.3.2 δ-Gromov-Hausdorff convergence 224

8.3.3 Pre-compactness of Finsler manifolds 231

8.3.4 On the Gauss-Bonnet-Chern theorem 233

Exercises 235

9 Minimal Immersions and Harmonic Maps 237

9.1 Isometric immersions 237

9.1.1 Finsler submanifolds 237

9.1.2 The variation of the volume 241

9.1.3 Non-existence of compact minimal submanifolds 245

9.2 Rigidity of minimal submanifolds 246

9.2.1 Minimal surfaces in Minkowski spaces 246

9.2.2 Minimal surfaces in (α, β)-spaces 249

9.2.3 Minimal surfaces in special Minkowskian (α, β)-spaces 253

9.3 Harmonic maps 259

9.3.1 A divergence formula 259

9.3.2 Harmonic maps 262

9.3.3 Composition maps 267

9.4 Second variation of harmonic maps 270

9.4.1 The second variation 270

9.4.2 Stress-energy tensor 273

9.5 Harmonic maps between complex Finsler manifolds 276

9.5.1 Complex Finsler manifolds 276

9.5.2 Harmonic maps between complex Finsler manifolds 280

9.5.3 Holomorphic maps 287

Exercises 289

10 Einstein Metrics 293

10.1 Projective rigidity and m-th root metrics 293

10.1.1 Projective rigidity of Einstein metrics 293

10.1.2 m-th root Einstein metrics 296

10.2 The Ricci rigidity and Douglas-Einstein metrics 298

10.2.1 The Ricci rigidity 298

10.2.2 Douglas (α, β)-metrics 300

10.3 Einstein (α, β)-metrics 305

10.3.1 Polynomial (α, β)-metrics 305

10.3.2 Kropina metrics 307

Exercises 317

11 Miscellaneous Topics 319

11.1 Conformal changes 319

11.1.1 Conformal changes 319

11.1.2 Conformally flat metrics 322

11.1.3 Conformally flat (α, β)-metrics 328

11.2 Conformally vector fields 330

11.2.1 Conformal vector fields 330

11.2.2 Conformal vector fields on a Randers manifold 338

11.3 A class of critical Finsler metrics 344

11.3.1 The Einstein-Hilbert functional 344

11.3.2 Some special (ε)-critical metrics 349

11.4 The first eigenvalue of Finsler Laplacian and the generalized maximal principle 351

11.4.1 Finsler Laplacian and weighted Ricci curvature 351

11.4.2 Lichnerowicz-Obata estimates 354

11.4.3 Li-Yau-Zhong-Yang type estimates 360

11.4.4 Mckean type estimates 364

Exercises 365

Appendix A Maple Program 367

A.1 Spray coefficients of two-dimensional Finsler metrics 367

A.2 Gauss curvature 372

A.3 Spray coefficients of (α, β)-metrics 374

Bibliography 379

Index 389