The subject matter of this book had its genesis in Riemann's 1854 "habil-itation" address: "Uber die Hypothesen, welche der Geometrie zu Grundeliegen" (On the Hypotheses, which lie at the Foundations of Geometry).Volume II of Spivak's Differential Geometry contains an English translationof this influential lecture, with a commentary by Spivak himself. Riemann, undoubtedly the greatest mathematician of the 19th century,aimed at introducing the notion of a manifold and its structures. The prob-lem involved great difficulties. But, with hypotheses on the smoothness ofthe functions in question, the issues can be settled satisfactorily and thereis now a complete treatment. Traditionally, the structure being focused on is the Riemannian metric,which is a quadratic differential form. Put another way, it is a smoothlyvarying family of inner products, one on each tangent space.

Preface

Acknowledgments

PART ONE Finsler Manifolds and Their Curvature

CHAPTER 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. Enler'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

 Referenees forCHAPTER 1

CHAPTER 2 The Chern Connection

 2.0 Prologue

 2.1 The Vector Bundleand π*TM and Related Objects

 2.2 Coordinate Bases Versus Special Orthonormal Bases

 2.3 The Nonlinear Connection on the Manifold TM \\O

 2.4 The Chern Connection on π*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 forCHAPTER 2

CHAPTER 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 dxk Λ dxl Terms

3.4 B. Consequences from the dxk Λ 1/Fδyl Terms

3.4 C. Consequences from the 1/Fδ Λ yk 1/Fδyl Terms

 3.5 Second Bianchi Identities

 3.6 Interchange Formulas or Ricci Identities

 3.7 Lie Brackets among the δ/δx and the F■/■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 forCHAPTER 3

CHAPTER 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 forCHAPTER 4

PART TWO Calculus of Variations and Comparison Theorems

CHAPTER 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 forCHAPTER 5

CHAPTER 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 Geodesics

 6.6 The Hopf-Rinow Theorem

 References forCHAPTER 6

CHAPTER 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. Geodesics 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 forCHAPTER 7

CHAPTER 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 WhichandDepend on y

 8.5 Generic Properties of the Cut Locus

 8.6 Additional Properties of Cutx When M Is Compact

 8.7 Shortest Geodesics within Homotopy Classes

 8.8 Synge's Theorem

 References forCHAPTER 8

CHAPTER 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 forCHAPTER 9

PART THREE Special Finsler Spaces over the Reals

CHAPTER 10 Berwald Spaces and Szabo'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 Szabo'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

CHAPTER 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

CHAPTER 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 E and E

 12.3 Growth Rates of Solutions of E + λ 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 Poincard 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

CHAPTER 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

 13.3 Warped 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 Weitzenbock Formula and the Bochner Technique

 References for Chapter 13

CHAPTER 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-Santal6 Inequality

 14.8 The Legendre Transformation

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

 References for Chapter 14

Bibliography

Index