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內容簡介

　　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.

內頁插圖

目錄

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）60：幾何VI 黎曼幾何 下載 mobi epub pdf txt 電子書

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