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內容簡介

　　Since the early work of Gauss and Riemann， differential geometry has grown into a vast network of ideas and approaches， encompassing local considerations such as differential invariants and jets as well as global ideas， such as Morse theory and characteristic classes： In this volume of the Encyclopaedia， the authors give a tour of the principal areas and methods of modern differential geometry. The book is structured so that the reader may choose parts of the text to read and still take away a completed picture ofsome area ofdifferential geometry Beginning at the introductory level with curves in Euclidean
　　space， the sections become more challenging. arriving finally at the advanced topics which form the greatest part of the book：transformation groups. the geometry of differential equations，geometric structures， the equivalence problem the geometry ofelliptic operators， G-structures and contact geometry. As an overview of the major current methods of differential geometry， EMS 28 is a map of these different ideas which explains the interesting points at every
　　stop， The authors' intention is that the reader should gain a new understanding of geometry from the process of reading this survey.

內頁插圖

目錄

Preface
Chaptcr 1．Introduction：A Metamathematical View of Differential Geometry
1．Algebra and Geometry—theDuality of the Intellect
2．Two Examples：Algebraic Geometry，Propositional Logic and Set Theory
3．On the History of Geometry
4．Differential Calculus and Commutative Algebra
5．What is Differential Geometry？

Chapter2．The Geometry of Surfaces
1．Curves in Euclidean Space
1．1．Curves
1．2．The Natural Parametrization and the intrinsic Geometry of Curves
1．3．Curvature．The Frenet Frame
1．4．Affine and Unimodular Properties of Curves
2．Surfaces in E3
2．1．Surfaces Charts
2．2．The First Quadratic Form．The Intrinsic Geometry of a Surface
2．3．The Second Quadratic Form．The Extrinsic Geometry of a Surface
2．4．Derivation Formulae．The First and Second Quadratic Forms
2．5．The Geodesic Curvature of Curves Geodesics
2．6．Parallel Transport of Tangent Vectors on a Surface．Covariant Differentiation．Connection 2．7．Deficiencies of Loops，the“Theorema Egregium”of Gauss and the Gauss—Bonnet Formula 2．8．The Link Between the First and Second Quadratic Forms．
The Gauss Equation and the Peterson—Mainardi—Codazzi Equations
2．9．The Moving Frame Method in the Theory of Surfaces
2．10．A Complete System of lnvariants of a Surface
3．Multidimensional Surfaces
3．1．n—Dimensional Surfaces in En+p．
3．2．Covariant Differentiation and the Second Quadratic Form
3．3．Normal Connection on a Surface．The Derivation Formulae
3．4．The Multidimensional Version of the Gauss—Peterson Mainardi—Codazzi Equations．Ricci’sTheorem 3．5．The Geometrical Meaning and Algebraic Properties of the Curvature Tensor 3．6．Hypersurfaces．Mean Curvatures．The Fonnulae of Steiner and Weyl 3．7．Rigidity of Multidimensional Surfaces

Chapter 3．The Field Approach of Riemann
1．From the Intrinsic Geometry of Gauss to Riemannian Geometrv
1．1．The Essence of Riemann’s Approach
1．2．Intrinsic Description of Surfaces
1．3．The Field Point of View on Geometry
1．4．Two Examples
2．Manifolds and Bundles（the BasicConcepts）
2．1 Why Do We Need Manifolds？
2．2．Definition of a Manifold
2．3．The Category of Smooth Manifolds
2．4．Smooth Bundles
3．Tensor Fields and Differential Forms
3．1．Tangent Vectors
3．2．The Tangent Bundle and Vector Fields
3．3 Covectors，the Cotangent Bundle and Differential Forms of the First Degree 3．4．Tensors and Tensor Fields
3．5．The Behaviour of Tensor Fields Under Maps．The Lie Derivative
3．6．The Exterior Differential．The de Rham Complex
4．Riemannian Manifolds and Manifolds with a Linear COnnectiOn
4．1．Riemannian Metric
4．2．Construction of Riemannian Metrics
4．3．Linear Connections
4．4．Normal Coordinates
4．5．A Riemannian Manifold as a Metric Space Completeness
4．6．Curvature
4．7．The Algebraic Structure of the Curvature Tensor．The Ricci and Weyl Tensors and Scalar Curvature
4．8．Sectional Curvature．Spaces of Constant Curvature
4．9．The Holonomy Group and the de Rham Decomposition
4．10．The Berger—Claass—ification of Holonomy Groups·Kahler and Quaternion Manifolds．
5．The Geometry of Symbols
5．1．Differential Operators in Bundles
5．2．Symbols of Differential Operators
5．3．Connections and Quantization．
5．4．Poisson Bracketsand Hamiltonian Formalism
5．5．Poissonian and Symplectic Structures
5．6．Left．Invariant Hamiltonian Formalism on Lie Groups

Chapter 4．The Group Approach of Lie and Klein．The Geometry of Transformation Groups．
1．Symmetries in Geometry
1．1．Symmetries and Groups
1．2．Symmetry and Integrability
1．3．KIein’S Erlangen Programme．
2．Homogeneous Spaces
2．1．Lie Groups
2．2．The Action ofthe Lie Group on a Manifold
2．3．Correspondence Between Lie Groups and Lie Algebras
2．4．Infinitesimal Description of Homogeneous Spaces
2．5．The Isotropy Representation．Order of a Homogeneous Space
2．6．The Principle of Extension．Invariant Tensor Fields on Homogeneous Spaces
2．7．Primitive and Imprimitive Actions
3．Invariant Connections on a Homogeneous Space
3．1．A General Description
3．2．Reductive Homogeneous Spaces
3．3．Atline Symmetric Spaces
4．Homogeneous Riemannian Manifolds
4．1．Infinitesimal Description
4．2．Thc Link Between Curvature and the Structure of the GrouP of Motions
4．3．Naturally Reductive Spaces
4．4．Symmetric Riemannian Spaces
4．5．Holonomy Groups of Homogeneous Riemannian Manifolds
Kahlerian and Quaternion Homogeneous Spaces
5．Homogeneous Symplectic Manifolds
5．1．Motivation and Definitions
5．2．Examoles
5．3．Homogeneous Hamiltonian Manifolds
5．4．Homogeneous Symplectic Manifolds and Affine Actions

Chapter 5．The Geometry of Differential Equations
1．Elementary Geometry of a First—Order Differential Equation
1．1 Ordinary Differential Equations
1．2．The General Case．
1．3．Geometrical Integration
2．Contact Geometry and Lie’s Theory of First．Order Equations
2．1．Contact Structure on J1
2．2．Generalized Solutions and Integral Manifolds ofthe Contact Structure 2．3 Contact Transformations
2．4．Contact Vector Fields
2．5 The Cauchy Problem
2．6．Symmetries．Local Equivalence
3．The Geometry ofDistributions
3．1 Distributions
3．2．A Distribution of Codimension I．The Theorem Of DarbOux．
3．3．Involutive Systems of Equations
3．4．The Intrinsic and Extrinsic Geometrv of First_Order Differential Equations 4．Spaces ofJets and Differential Equations
4．1．Jets．
4．2．The Caftan Distribution
4．3 Lie Transformations
4．4 Intrinsic and ExtrinsicGeometries
5．The Theory of Compatibility and Formal Integrabilitv
5．1．Prolongations ofDifferential Equations
5．2．Formal Integrability
5．3．Symbols
5．4．The Spencer δ—Cohomology
5．5．Involutivity
6．Cartan’S Theory of Systems in Involution
6．1 PolarSystems，Characters and Genres
6．2．Involutivity and Cartan’S Existence Theorems
7．The Geometry of Infinitely Prolonged Equations
7．1．What is a Differential Equation？
7．2．Infinitely Prolonged Equations
7．3．C—Maps and Higher Symmetries

Chapter 6．Geometric Structures
1．GeometricQuantities and Geometric Structures
1．1 What is a Geometric Quantity？
1．2．Bundles of Frames and Coframes
1．3．Geometric Quantities（Structures）as Equivariant Functions
on the Manifold of Coframes
1．4．Examples．Infinitesimally Homogeneous Geometric Structures
1．5．Natural Geometric Structures and the Principle of Covanance
……
Chapter7．The Equivalence Problem，Differential Invariants and Pseudogroups
Chapter8．Global Aspects of Differential Geometry
Commentary on the References
References
Author Index
Subject Index

前言/序言

　　要使我國的數學事業更好地發展起來，需要數學傢淡泊名利並付齣更艱苦地努力。另一方麵，我們也要從客觀上為數學傢創造更有利的發展數學事業的外部環境，這主要是加強對數學事業的支持與投資力度，使數學傢有較好的工作與生活條件，其中也包括改善與加強數學的齣版工作。
　　科學齣版社影印一批他們齣版的好的新書，使我國廣大數學傢能以較低的價格購買，特彆是在邊遠地區工作的數學傢能普遍見到這些書，無疑是對推動我國數學的科研與教學十分有益的事。
　　這次科學齣版社購買瞭版權，一次影印瞭23本施普林格齣版社齣版的數學書，就是一件好事，也是值得繼續做下去的事情。大體上分一下，這23本書中，包括基礎數學書5本，應用數學書6本與計算數學書12本，其中有些書也具有交叉性質。這些書都是很新的，2000年以後齣版的占絕大部分，共計16本，其餘的也是1990年以後齣版的。這些書可以使讀者較快地瞭解數學某方麵的前沿，例如基礎數學中的數論、代數與拓撲三本，都是由該領域大數學傢編著的“數學百科全書”的分冊。對從事這方麵研究的數學傢瞭解該領域的前沿與全貌很有幫助。按照學科的特點，基礎數學類的書以“經典”為主，應用和計算數學類的書以“前沿”為主。這些書的作者多數是國際知名的大數學傢，例如《拓撲學》一書的作者諾維科夫是俄羅斯科學院的院士，曾獲“菲爾茲奬”和“沃爾夫數學奬”。這些大數學傢的著作無疑將會對我國的科研人員起到非常好的指導作用。
　　當然，23本書隻能涵蓋數學的一部分，所以，這項工作還應該繼續做下去。更進一步，有些讀者麵較廣的好書還應該翻譯成中文齣版，使之有更大的讀者群。
　　總之，我對科學齣版社影印施普林格齣版社的部分數學著作這一舉措錶示熱烈的支持，並盼望這一工作取得更大的成績。
國外數學名著係列（續一 影印版）55：幾何I 微分幾何基本思想與概念 [Geometry 1 Basic Ideas and Concepts of Differential Geometry] 下載 mobi epub pdf txt 電子書

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