黎曼-芬斯勒几何导论

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. Enlers 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 for Chapter 1

CHAPTER 2

The Chern Connection

2.0 Prologue

2.1 The Vector Bundleand 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

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

CHAPTER 3

Curvature and Schurs Lemma

3.1Conventions and the hh-， hv-， w-curvatures

3.2First Bianchi Identities from Torsion Freeness

3.3Formulas for R and P in Natural Coordinates

3.4First Bianchi Identities from "Almost" g-compatibility

3.4 A. Consequences from the

3.4 B. Consequences from the

3.4 C. Consequences from the

3.5Second Bianchi Identities

3.6Interchange Formulas or Ricci Identities

3.7Lie Brackets among the

Oy

3.8Derivatives of the Geodesic Spray Coefficients

3.9The Flag Curvature

3.9 A. Its Definition and Its Predecessor

3.9 B. An Interesting Family of Examples of Numata Type

3.10 Schurs Lemma

References for Chapter 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. Runds 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

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 Curvatures Sign Influences Geodesic Rays

References for Chapter 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 for Chapter 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. TheRicci 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

CHAPTER 8

The Cut and Conjugate Loci， and Synges 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 Synges Theorem

References for Chapter 8

CHAPTER 9

The Cartan-Hadamard Theorem and

Rauchs 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 Rauchs 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 Rauchs First Comparison Theorem

9.7 Jacobi Fields on Space Forms

9.8 Applications of Rauchs Theorem

References for Chapter 9

The subject matter of this book had its genesis in Riemanns 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 Spivaks 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. The resultinggeometry —— Riemannian geometry —— has undergone tremendous develop-ment in this century. Areas in which it has had significant impact includeEinsteins theory of general relativity， and global differential geometry. In the context of Riemanns lecture， this restriction to a quadratic dif-ferential form