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　　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.

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目录

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

　　要使我国的数学事业更好地发展起来，需要数学家淡泊名利并付出更艰苦地努力。另一方面，我们也要从客观上为数学家创造更有利的发展数学事业的外部环境，这主要是加强对数学事业的支持与投资力度，使数学家有较好的工作与生活条件，其中也包括改善与加强数学的出版工作。
　　科学出版社影印一批他们出版的好的新书，使我国广大数学家能以较低的价格购买，特别是在边远地区工作的数学家能普遍见到这些书，无疑是对推动我国数学的科研与教学十分有益的事。
　　这次科学出版社购买了版权，一次影印了23本施普林格出版社出版的数学书，就是一件好事，也是值得继续做下去的事情。大体上分一下，这23本书中，包括基础数学书5本，应用数学书6本与计算数学书12本，其中有些书也具有交叉性质。这些书都是很新的，2000年以后出版的占绝大部分，共计16本，其余的也是1990年以后出版的。这些书可以使读者较快地了解数学某方面的前沿，例如基础数学中的数论、代数与拓扑三本，都是由该领域大数学家编著的“数学百科全书”的分册。对从事这方面研究的数学家了解该领域的前沿与全貌很有帮助。按照学科的特点，基础数学类的书以“经典”为主，应用和计算数学类的书以“前沿”为主。这些书的作者多数是国际知名的大数学家，例如《拓扑学》一书的作者诺维科夫是俄罗斯科学院的院士，曾获“菲尔兹奖”和“沃尔夫数学奖”。这些大数学家的著作无疑将会对我国的科研人员起到非常好的指导作用。
　　当然，23本书只能涵盖数学的一部分，所以，这项工作还应该继续做下去。更进一步，有些读者面较广的好书还应该翻译成中文出版，使之有更大的读者群。
　　总之，我对科学出版社影印施普林格出版社的部分数学著作这一举措表示热烈的支持，并盼望这一工作取得更大的成绩。
国外数学名著系列（续一 影印版）55：几何I 微分几何基本思想与概念 [Geometry 1 Basic Ideas and Concepts of Differential Geometry] 电子书 下载 mobi epub pdf txt

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