国外数学名著系列（续一 影印版）60：几何VI 黎曼几何 下载 mobi epub pdf 电子书 2024

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　　This book treats that part of Riemannian geometry related to more classical topics in a very original， clear and solid style. Before going to Riemannian geometry， the author presents a more general theory of manifolds with a linear connection. Having in mind different generalizations of Riemannian manifolds， it is clearly stressed which notions and theorems belong to Riemannian geometry and which of them are of a more general nature. Much attention is paid to
　　transformation groups of smooth manifolds.Throughout the book， different aspects of symmetric spaces are treated The author successfully combines the co-ordinate and invariant approaches to differential geometry， which give the reader tools for practical calculations as well as a theoretical understanding of the subject. The book contains a very usefullarge appendix on foundations of differentiable manifolds and basic structures on them which makes it self contained and practically independent from other sources.
　　The results are well presented and useful for students in mathematics and theoretical physics， and for experts in these fields.The book can serve as a textbook for students doing geometry， as well as a reference book for professional mathematicians and physicists.

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目录

Preface
Chapter 1．Afne Connections
1．Connection on a Manifold
2．Covariant Differentiation and Parallel Translation Along a Curve
3．Geodesics
4．Exponential Mapping and Normal Neighborhoods
5．Whitehead Theorem
6．Normal Convex Neighborhoods
7．Existence of Leray Coverings

Chapter 2．Covariant Differentiation．Curvature
1．Covariant Differentiation
2．The Case of Tensors of Type
3．Torsion Tensor and Symmetric Connections
4．Geometric Meaning of the Symmetry of a Connection
5．Commutativity of Second Covariant Derivatives
6．Curvature Tensor of an Afne Connection
7．Space with Absolute Parallelism
8．Bianci Identities
9．Trace of the Curvature Tensor
10．Ricci Tensor

Chapter 3．Affine Mappings．Submanifolds
1．Afne Mappings
2．Affinities
3．Afne Coverings
4．Restriction of a Connection to a Submanifold
5．Induced Connection on a Normalized Submanifold
6．Gauss Formula and the Second Fundamental Form of a Normalized Submanifold
7．Totally Geodesic and Auto—Parallel Submanifolds
8．Normal Connection and the Weingarten Formula
9．Van der Waerden—Bortolotti Connection

Chapter 4．Structural Equations．Local Symmetries
1．Torsion and Curvature Forms
2．Caftan Structural Equations in Polar Coordinates
3．Existence of Afne Local Mappings
4．Locally Symmetric Afne Connection Spaces
5．Local Geodesic Symmetries
6．Semisymmetric Spaces

Chapter 5．Symmetric Spaces
1．Globally Symmetric Spaces
2．Germs of Smooth Mappings
3．Extensions of Affine Mappings
4．Uniqueness Theorem
5．Reduction of Locally Symmetric Spaces to Globally Symmetric Spaces
6．Properties of Symmetries in Globally Symmetric Spaces
7．Symmetric Spaces
8．Examples of Symmetric Spaces
9．Coincidence of Classes of Symmetric and Globally Symmetric Spaces

Chapter 6．Connections on Lie Groups
1．Invariant Construction of the Canonical Connection
2．Morphisms of Symmetric Spaces as Affine Mappings
3．Left—Invariant Connections on a Lie Group
4．Cartan Connections
5．Left Cartan Connection
6．Right—Invariant Vector Fields
7．Right Cartan Connection

Chapter 7．Lie Functor
1．Categories
2．Functors
3．Lie Functor
4．Kernel and Image of a Lie Group Homomorphism
5．Campbell—Hausdorff Theorem
6．Dynkin Polynomials
7．Local Lie Groups
8．Bijectivity of the Lie Functor

Chapter 8．Affine Fields and Related Topics
1．Affine Fields
2．Dimension of the Lie Algebra of Affine Fields
3．Completeness of Affine Fields
4．Mappings of Left and Right Translation on a Symmetric Space
5．Derivations on Manifolds with Multiplication
6．Lie Algebra of Derivations
7．Involutive Automorphism of the Derivation Algebra of a Symmetric Space
8．Symmetric Algebras and Lie Ternaries
9．Lie Ternary of a Symmetric Space

Chapter 9．Cartan Theorem
1．Functor s
2．Comparison of the Functor s with the Lie Functor
3．Properties of the Functor s
4．Computation of the Lie Ternary of the Space
5．Fundamental Group of the Quotient Space
6．Symmetric Space with a Given Lie Ternary
7．Coverings
8．Cartan Theorem
9．Identification of Homogeneous Spaces with Quotient Spaces
10．Trauslations of a Symmetric Space
11．Proof of the Cartan Theorem

Chapter 10．Palais and Kobayashi Theorems
1．Infinite—Dimensional Manifolds and Lie Groups
2．Vector Fields Induced by a Lie Group Action
3．Palais Theorem
4．Kobayashi Theorem
5．Affine Automorphism Group
6．Automorphism Group of a Symmetric Space
7．Translation Group of a Symmetric Space

Chapter 11．Lagrangians in Riemannian Spaces
1．Riemannian and Pseudo—Riemannian Spaces
2．Riemannian Connections
3．Geodesics in a Riemannian Space
4．Simplest Problem of the Calculus of Variations
5．Euler—Lagrange Equations
6．Minimum Curves and Extremals
7．Regular Lagrangians
8．Extremals of the Energy Lagrangian

Chapter 12．Metric Properties of Geodesics
1．Length of a Curve in a Riemannian Space
2．Natural Parameter
3．Riemannian Distance and Shortest Arcs
4．Extremals of the Length Lagrangian
5．Riemannian Coordinates
6．Gauss Lemma
7．Geodesics are Locally Shortest Arcs
8．Smoothness of Shortest Arcs
9．Local Existence of Shortest Arcs
10．Intrinsic Metric
11．Hopf—Rinow Theorem

Chapter 13．Harmonic Functionals and Related Topics
1．Riemannian Volume Element
2．Discriminant Tensor
3．Foss—Weyl Formula
4．Case n=2
5．Laplace Operator on a Riemannian Space
6．The Green Formulas
7．Existence of Harmonic Functions with a Nonzero Differential
8．Conjugate Harmonic Functions
9．Isothermal Coordinates
10．Semi—Cartesian Coordinates
11．Cartesian Coordinates
……
Chapter 14．Minimal Surfaces
Chapter 15．Curvature in Riemannian Space
Chapter 16．Gaussian Curvature
Chapter 17．Some Special Tensors
Chapter 18．Surfaces with Conformal Structure
Chapter 19．Mappings and Submanifolds Ⅰ
Chapter 20．Submanifolds Ⅱ
Chapter 21．Fundamental Forms of a Hypersurface
Chapter 22．Spaces of Constant Curvature
Chapter 23．Space Forms
Chapter 24．Four—Dimensional Manifolds
Chapter 25．Metrics on a Lie Group Ⅰ
Chapter 26．Metrics on a Lie Group Ⅱ
Chapter 27．Jacobi Theory
Chapter 28．Some Additional Theorems Ⅰ
Chapter 29．Some Additional Theorems Ⅱ
Chapter 30．Smooth Manifolds
Chapter 31．Tangent Vectors
Chapter 32．Submanifolds of a Smooth Manifold
Chapter 33．Vector and Tensor Fields．Differential Forms
Chapter 34．Vector Bundles
Chapter 35．Connections on Vector Bundles
Chapter 36．Curvature Tensor
Suggested Reading
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）60：几何VI 黎曼几何 下载 mobi epub pdf txt 电子书 格式

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