An Introduction to Riemann-Finsler Geometry

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1 D. Bao S.-S. Chern Z. Shen An Introduction to Riemann-Finsler Geometry With 20 Illustrations Springer

2 Contents Preface Acknowledgments vn xiii PART ONE Finsler Manifolds and Their Curvature CHAPTER 1 Finsler Manifolds and the Fundamentals of Minkowski Norms Physical Motivations Finsler Structures: Definitions and Conventions Two Basic Properties of Minkowski Norms A. Euler's Theorem B. A Fundamental Inequality C. Interpretations of the Fundamental Inequality Explicit Examples of Finsler Manifolds A. Minkowski and Locally Minkowski Spaces B. Riemannian Manifolds C. Randers Spaces D. Berwald Spaces E. Finsler Spaces of Constant Flag Curvature The Fundamental Tensor and the Cartan Tensor 22 * References for Chapter 1 25 CHAPTER 2 The Chern Connection Prologue The Vector Bundle TT*TM and Related Objects Coordinate Bases Versus Special Orthonormal Bases The Nonlinear Connection on the Manifold TM \ The Chern Connection on -K*TM 37

3 xvi Contents 2.5 Index Gymnastics A. The Slash (...) s and the Semicolon (...) ;s B. Covariant Derivatives of the Fundamental Tensor g C. Covariant Derivatives of the Distinguished 46 * References for Chapter 2 48 CHAPTER 3 Curvature and Schur's Lemma Conventions and the hh-, hv-, uu-curvatures First Bianchi Identities from Torsion Freeness Formulas for R and P in Natural Coordinates First Bianchi Identities from "Almost" ^-compatibility A. Consequences from the dx k A dx l Terms B. Consequences from the dx k A j?6y l Terms C. Consequences from the j^sy k A jsy 1 Terms Second Bianchi Identities Interchange Formulas or Ricci Identities Lie Brackets among the ~ and the F- ^ Derivatives of the Geodesic Spray Coefficients G x The Flag Curvature A. Its Definition and Its Predecessor B. An Interesting Family of Examples of Numata Type Schur's Lemma 75 * References for Chapter 3 80 CHAPTER 4 Finsler Surfaces and a Generalized Gauss Bonnet Theorem Prologue Minkowski Planes and a Useful Basis A. Rund's Differential Equation and Its Consequence B. A Criterion for Checking Strong Convexity The Equivalence Problem for Minkowski Planes The Berwald Frame and Our Geometrical Setup on SM The Chern Connection and the Invariants I, J, K The Riemannian Arc Length of the Indicatrix A Gauss-Bonnet Theorem for Landsberg Surfaces 105 * References for Chapter 4 110

4 Contents PART TWO Calculus of Variations and Comparison Theorems 111 CHAPTER 5 Variations of Arc Length, Jacobi Fields, the Effect of Curvature 5.1 The First Variation of Arc Length Ill 5.2 The Second Variation of Arc Length Geodesies and the Exponential Map Jacobi Fields How the Flag Curvature's Sign Influences Geodesic Rays 135 * References for Chapter Ill CHAPTER 6 The Gauss Lemma and the Hopf-Rinow Theorem The Gauss Lemma A. The Gauss Lemma Proper B. An Alternative Form of the Lemma C. Is the Exponential Map Ever a Local Isometry? Finsler Manifolds and Metric Spaces A. A Useful Technical Lemma B. Forward Metric Balls and Metric Spheres C. The Manifold Topology Versus the Metric Topology D. Forward Cauchy Sequences, Forward Completeness Short Geodesies Are Minimizing The Smoothness of Distance Functions A. On Minkowski Spaces B. On Finsler Manifolds Long Minimizing Geodesies The Hopf-Rinow Theorem 168 * References for Chapter CHAPTER 7 The Index Form and the Bonnet-Myers Theorem Conjugate Points The Index Form What Happens in the Absence of Conjugate Points? A. Geodesies Are Shortest Among "Nearby" Curves B. A Basic Index Lemma What Happens If Conjugate Points Are Present? The Cut Point Versus the First Conjugate Point 186

5 xviii Contents 7.6 Ricci Curvatures A. The Ricci Scalar Ric and the Ricci Tensor Ricij B. The Interplay between Ric and Ric tj The Bonnet-Myers Theorem 194 * References for Chapter CHAPTER 8 The Cut and Conjugate Loci, and Synge's Theorem Definitions The Cut Point and the First Conjugate Point Some Consequences of the Inverse Function Theorem The Manner in Which Cy and i y Depend on y Generic Properties of the Cut Locus Cut x Additional Properties of Cut x When M Is Compact Shortest Geodesies within Homotopy Classes Synge's Theorem 221 * References for Chapter CHAPTER 9 The Cartan Hadamard Theorem and Rauch's First Theorem Estimating the Growth of Jacobi Fields When Do Local Diffeomorphisms Become Covering Maps? Some Consequences of the Covering Homotopy Theorem The Cartan-Hadamard Theorem Prelude to Rauch's Theorem A. Transplanting Vector Fields B. A Second Basic Property of the Index Form C. Flag Curvature Versus Conjugate Points Rauch's First Comparison Theorem Jacobi Fields on Space Forms Applications of Rauch's Theorem 253 * References for Chapter 9 256

6 Contents xix PART THREE Special Finsler Spaces over the Reals 257 CHAPTER 10 Berwald Spaces and Szabo's Theorem for Berwald Surfaces Prologue Berwald Spaces Various Characterizations of Berwald Spaces Examples of Berwald Spaces A Fact about Flat Linear Connections Characterizing Locally Minkowski Spaces by Curvature Szabo's Rigidity Theorem for Berwald Surfaces A. The Theorem and Its Proof B. Distinguishing between y-local and y-global 279 * References for Chapter CHAPTER 11 Randers Spaces and an Elegant Theorem The Importance of Randers Spaces Randers Spaces, Positivity, and Strong Convexity A Matrix Result and Its Consequences The Geodesic Spray Coefficients of a Randers Metric The Nonlinear Connection for Randers Spaces A Useful and Elegant Theorem The Construction of y-global Berwald Spaces A. The Algorithm B. An Explicit Example in Three Dimensions 306 * References for Chapter CHAPTER 12 Constant Flag Curvature Spaces and Akbar-Zadeh's Theorem Prologue Characterizations of Constant Flag Curvature Useful Interpretations of E and E Growth Rates of Solutions of E + A E = Akbar-Zadeh's Rigidity Theorem Formulas for Machine Computations of K A. The Geodesic Spray Coefficients B. The Predecessor of the Flag Curvature 330

7 xx Contents 12.5 C. Maple Codes for the Gaussian Curvature A Poincare Disc That Is Only Forward Complete A. The Example and Its Yasuda-Shimada Pedigree B. The Finsler Function and Its Gaussian Curvature C. Geodesies; Forward and Backward Metric Discs D. Consistency with Akbar-Zadeh's Rigidity Theorem Non-Riemannian Projectively Flat S 2 with K \ A. Bryant's 2-parameter Family of Finsler Structures B. A Specific Finsler Metric from That Family 345 * References for Chapter CHAPTER 13 Riemannian Manifolds and Two of Hopf's Theorems The Levi-Civita (Christoffel) Connection Curvature A. Symmetries, Bianchi Identities, the Ricci Identity B. Sectional Curvature C. Ricci Curvature and Einstein Metrics Warped Products and Riemannian Space Forms A. One Special Class of Warped Products B. Spheres and Spaces of Constant Curvature C. Standard Models of Riemannian Space Forms Hopf's Classification of Riemannian Space Forms The Divergence Lemma and Hopf's Theorem The Weitzenbock Formula and the Bochner Technique 378 * References for Chapter CHAPTER 14 Minkowski Spaces, the Theorems of Deicke and Brickell Generalities and Examples The Riemannian Curvature of Each Minkowski Space The Riemannian Laplacian in Spherical Coordinates Deicke's Theorem The Extrinsic Curvature of the Level Spheres of F The Gauss Equations The Blaschke-Santalo Inequality The Legendre Transformation A Mixed-Volume Inequality, and Brickell's Theorem 412 * References for Chapter Bibliography 419 Index 427

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