黎曼-芬斯勒幾何導(dǎo)論

前言

　　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 constitutes only a special case. Nevertheless， Riemann sawthe great merit of this special case， so much so that he introduced for itthe curvature tensor and the notion of sectional curvature. Such was donethrough a Taylor expansion of the Riemannian metric.　The Riemann curvature tensor plays a major role in a fundamental prob-lem. Namely： how does one decide， in principle， whether two given Rie-mannian structures differ only by a coordinate transformation？ This wassolved in 1870， independently by Christoffel and Lipschitz， using differentmethods and without the benefit of tensor calculus. It was almost 50 yearslater， in 1917， that Levi-Civita introduced his notion of parallelism （equiv-alent to a connection）， thereby giving the solution a simple geometricalinterpretation.

內(nèi)容概要

　　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

書(shū)籍目錄

PrefaceAcknowledgmentsPART 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. 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　CHAPTER 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 　　2.4 The Chern Connection on TM    2.5 Index Gymnastics    References for Chapter 2　CHAPTER 3 Curvature and Schur's Lemma　  3.1  Conventions and the hh-, hv-, w-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　  3.5  Second Bianchi Identities  　3.6  Interchange Formulas or Ricci Identities　  3.7  Lie Brackets among the  and the   　3.8  Derivatives of the Geodesic Spray Coefficients Gi　  3.9  The Flag Curvature　  3.10 Schur's 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　  4.2 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,　CHAPTER 6 The Gauss Lemma and the Hopf-Rinow Theorem　CHAPTER 7 The Index Form and the Bonnet-Myers Theorem   CHAPTER 8 The Cut and Conjugate Loci, and Synge's Theorem  CHAPTER 9 The Cartan-Hadamard Theorem and Rauch's First TheoremPART THREE Special Finsler Spaces over the Reals  CHAPTER 10 Berwald Spaces and Szab6's Theorem for Berwald Surfaces  CHAPTER 11 Randers Spaces and an Elegant Theorem  CHAPTER 12 Constant Flag Curvature Spaces and Akbar-Zadeh's Theorem  CHAPTER 13 Riemannian Manifolds and Two of Hopf's Theorems  CHAPTER 14 Minkowski Spaces, the Theorems of Deicke and Brickell BibliographyIndex

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