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This book focuses on the elementary but essential problems in Riemann-Finsler Geometry, which include a repertoire of rigidity and comparison theorems, and an array of explicit examples, illustrating many phenomena which admit only Finslerian interpretations. "This book offers the most modern treatment of the topic ..." EMS Newsletter.

An Introduction to Riemann-Finsler Geometry

One Finsler Manifolds and Their Curvature.- 1 Finsler Manifolds and the Fundamentals of Minkowski Norms.- 1.0 Physical Motivations.- 1.1 Finsler Structures: Definitions and Conventions.- 1.2 Two Basic Properties of Minkowski Norms.- 1.2 A. Euler's Theorem.- 1.2 B. A Fundamental Inequality.- 1.2 C. Interpretations of the Fundamental Inequality.- 1.3 Explicit Examples of Finsler Manifolds.- 1.3 A. Minkowski and Locally Minkowski Spaces.- 1.3 B. Riemannian Manifolds.- 1.3 C. Randers Spaces.- 1.3 D. Berwald Spaces.- 1.3 E. Finsler Spaces of Constant Flag Curvature.- 1.4 The Fundamental Tensor and the Cartan Tensor.- * References for Chapter 1.- 2 The Chern Connection.- 2.0 Prologue.- 2.1 The Vector Bundle ?*TM and Related Objects.- 2.2 Coordinate Bases Versus Special Orthonormal Bases.- 2.3 The Nonlinear Connection on the Manifold TM \0.- 2.4 The Chern Connection on ?*TM.- 2.5 Index Gymnastics.- 2.5 A. The Slash (...)s and the Semicolon (...);s.- 2.5 B. Covariant Derivatives of the Fundamental Tensor g.- 2.5 C. Covariant Derivatives of the Distinguished ?.- * References for Chapter 2.- 3 Curvature and Schur's Lemma.- 3.1 Conventions and the hh-, hv-, vv-curvatures.- 3.2 First Bianchi Identities from Torsion Freeness.- 3.3 Formulas for R and P in Natural Coordinates.- 3.4 First Bianchi Identities from "Almost" g-compatibility.- 3.4 A. Consequences from the $$dx^k \wedge dx^l$$ Terms.- 3.4 B. Consequences from the $$dx^k \wedge \frac{1}{F}\delta y^l$$ Terms.- 3.4 C. Consequences from the $$\frac{1}{F}\delta y^k \wedge \frac{1}{F}\delta y^l$$ Terms.- 3.5 Second Bianchi Identities.- 3.6 Interchange Formulas or Ricci Identities.- 3.7 Lie Brackets among the $$\frac{\delta }{{\delta x}}$$ and the $$F\frac{\partial }{{\partial y}}$$.- 3.8 Derivatives of the Geodesic Spray Coefficients Gi.- 3.9 The Flag Curvature.- 3.9 A. Its Definition and Its Predecessor.- 3.9 B. An Interesting Family of Examples of Numata Type.- 3.10 Schur's Lemma.- *References for Chapter 3.- 4 Finsler Surfaces and a Generalized Gauss-Bonnet Theorem.- 4.0 Prologue.- 4.1 Minkowski Planes and a Useful Basis.- 4.1 A. Rund's Differential Equation and Its Consequence.- 4.1 B. A Criterion for Checking Strong Convexity.- 4.2 The Equivalence Problem for Minkowski Planes.- 4.3 The Berwald Frame and Our Geometrical Setup on SM.- 4.4 The Chern Connection and the Invariants I, J, K.- 4.5 The Riemannian Arc Length of the Indicatrix.- 4.6 A Gauss-Bonnet Theorem for Landsberg Surfaces.- *References for Chapter 4.- Two Calculus of Variations and Comparison Theorems.- 5 Variations of Arc Length, Jacobi Fields, the Effect of Curvature.- 5.1 The First Variation of Arc Length.- 5.2 The Second Variation of Arc Length.- 5.3 Geodesics and the Exponential Map.- 5.4 Jacobi Fields.- 5.5 How the Flag Curvature's Sign Influences Geodesic Rays.- *References for Chapter 5.- 6 The Gauss Lemma and the Hopf-Rinow Theorem.- 6.1 The Gauss Lemma.- 6.1 A. The Gauss Lemma Proper.- 6.1 B. An Alternative Form of the Lemma.- 6.1 C. Is the Exponential Map Ever a Local Isometry?.- 6.2 Finsler Manifolds and Metric Spaces.- 6.2 A. A Useful Technical Lemma.- 6.2 B. Forward Metric Balls and Metric Spheres.- 6.2 C. The Manifold Topology Versus the Metric Topology.- 6.2 D. Forward Cauchy Sequences, Forward Completeness.- 6.3 Short Geodesics Are Minimizing.- 6.4 The Smoothness of Distance Functions.- 6.4 A. On Minkowski Spaces.- 6.4 B. On Finsler Manifolds.- 6.5 Long Minimizing Geodesies.- 6.6 The Hopf-Rinow Theorem.- *References for Chapter 6.- 7 The Index Form and the Bonnet-Myers Theorem.- 7.1 Conjugate Points.- 7.2 The Index Form.- 7.3 What Happens in the Absence of Conjugate Points?.- 7.3 A. Geodesies Are Shortest Among "Nearby" Curves.- 7.3 B. A Basic Index Lemma.- 7.4 What Happens If Conjugate Points Are Present?.- 7.5 The Cut Point Versus the First Conjugate Point.- 7.6 Ricci Curvatures.- 7.6 A. The Ricci Scalar Ric and the Ricci Tensor Ricij.- 7.6 B. The Interplay between Ric and RiCij.- 7.7 The Bonnet-Myers Theorem.- *References for Chapter 7.- 8 The Cut and Conjugate Loci, and Synge's Theorem.- 8.1 Definitions.- 8.2 The Cut Point and the First Conjugate Point.- 8.3 Some Consequences of the Inverse Function Theorem.- 8.4 The Manner in Which cy and iy Depend on y.- 8.5 Generic Properties of the Cut Locus Cutx.- 8.6 Additional Properties of Cutx When M Is Compact.- 8.7 Shortest Geodesics within Homotopy Classes.- 8.8 Synge's Theorem.- *References for Chapter 8.- 9 The Cartan-Hadamard Theorem and Rauch's First Theorem.- 9.1 Estimating the Growth of Jacobi Fields.- 9.2 When Do Local Diffeomorphisms Become Covering Maps?.- 9.3 Some Consequences of the Covering Homotopy Theorem.- 9.4 The Cartan-Hadamard Theorem.- 9.5 Prelude to Rauch's Theorem.- 9.5 A. Transplanting Vector Fields.- 9.5 B. A Second Basic Property of the Index Form.- 9.5 C. Flag Curvature Versus Conjugate Points.- 9.6 Rauch's First Comparison Theorem.- 9.7 Jacobi Fields on Space Forms.- 9.8 Applications of Rauch's Theorem.- *References for Chapter 9.- Three Special Finsler Spaces over the Reals.- 10 Berwald Spaces and Szabó's Theorem for Berwald Surfaces.- 10.0 Prologue.- 10.1 Berwald Spaces.- 10.2 Various Characterizations of Berwald Spaces.- 10.3 Examples of Berwald Spaces.- 10.4 A Fact about Flat Linear Connections.- 10.5 Characterizing Locally Minkowski Spaces by Curvature.- 10.6 Szabó's Rigidity Theorem for Berwald Surfaces.- 10.6 A. The Theorem and Its Proof.- 10.6 B. Distinguishing between y-local and y-global.- *References for Chapter 10.- 11 Randers Spaces and an Elegant Theorem.- 11.0 The Importance of Randers Spaces.- 11.1 Randers Spaces, Positivity, and Strong Convexity.- 11.2 A Matrix Result and Its Consequences.- 11.3 The Geodesic Spray Coefficients of a Randers Metric.- 11.4 The Nonlinear Connection for Randers Spaces.- 11.5 A Useful and Elegant Theorem.- 11.6 The Construction of y-global Berwald Spaces.- 11.6 A. The Algorithm.- 11.6 B. An Explicit Example in Three Dimensions.- *References for Chapter 11 309.- 12 Constant Flag Curvature Spaces and Akbar-Zadeh's Theorem.- 12.0 Prologue.- 12.1 Characterizations of Constant Flag Curvature.- 12.2 Useful Interpretations of ? and Ë.- 12.3 Growth Rates of Solutions of Ë + ? E = 0.- 12.4 Akbar-Zadeh's Rigidity Theorem.- 12.5 Formulas for Machine Computations of K.- 12.5 A. The Geodesic Spray Coefficients.- 12.5 B. The Predecessor of the Flag Curvature.- 12.5 C. Maple Codes for the Gaussian Curvature.- 12.6 A Poincaré Disc That Is Only Forward Complete.- 12.6 A. The Example and Its Yasuda-Shimada Pedigree.- 12.6 B. The Finsler Function and Its Gaussian Curvature.- 12.6 C. Geodesics; Forward and Backward Metric Discs.- 12.6 D. Consistency with Akbar-Zadeh's Rigidity Theorem.- 12.7 Non-Riemannian Projectively Flat S2 with K = 1.- 12.7 A. Bryant's 2-parameter Family of Finsler Structures.- 12.7 B. A Specific Finsler Metric from That Family.- *References for Chapter 12 350.- 13 Riemannian Manifolds and Two of Hopf's Theorems.- 13.1 The Levi-Civita (Christoffel) Connection.- 13.2 Curvature.- 13.2 A. Symmetries, Bianchi Identities, the Ricci Identity.- 13.2 B. Sectional Curvature.- 13.2 C. Ricci Curvature and Einstein Metrics.- 13.3Warped Products and Riemannian Space Forms.- 13.3 A. One Special Class of Warped Products.- 13.3 B. Spheres and Spaces of Constant Curvature.- 13.3 C. Standard Models of Riemannian Space Forms.- 13.4 Hopf's Classification of Riemannian Space Forms.- 13.5 The Divergence Lemma and Hopf's Theorem.- 13.6 The Weitzenböck Formula and the Bochner Technique.- *References for Chapter 13.- 14 Minkowski Spaces, the Theorems of Deicke and Brickell.- 14.1 Generalities and Examples.- 14.2 The Riemannian Curvature of Each Minkowski Space.- 14.3 The Riemannian Laplacian in Spherical Coordinates.- 14.4 Deicke's Theorem.- 14.5 The Extrinsic Curvature of the Level Spheres of F.- 14.6 The Gauss Equations.- 14.7 The Blaschke-Santaló Inequality.- 14.8 The Legendre Transformation.- 14.9 A Mixed-Volume Inequality, and Brickell's Theorem.- * References for Chapter 14.