几何VI-黎曼几何 本书特色
This book treats that part of Riemanniageometry related to more classical topics ia very original， clear and solid style. Before going to Riemanniageometry， the author presents a more general theory of manifolds with a linear connection. Having imind different generalizations of Riemanniamanifolds， it is clearly stressed which notions and theorems belong to Riemanniageometry and which of them are of a more general nature. Much attentiois paid to
transformatiogroups 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 ofoundations of differentiable manifolds and basic structures othem which makes it self contained and practically independent from other sources.
The results are well presented and useful for students imathematics and theoretical physics， and for experts ithese fields.The book caserve as a textbook for students doing geometry， as well as a reference book for professional mathematicians and physicists.
几何VI-黎曼几何 内容简介
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几何VI-黎曼几何 目录
Preface
Chapter 1．Afne Connections
1．Connectiooa Manifold
2．Covariant Differentiatioand Parallel TranslatioAlong 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．TorsioTensor and Symmetric Connections
4．Geometric Meaning of the Symmetry of a Connection
5．Commutativity of Second Covariant Derivatives
6．Curvature Tensor of aAfne 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．Restrictioof a Connectioto a Submanifold
5．Induced Connectiooa Normalized Submanifold
6．Gauss Formula and the Second Fundamental Form of a Normalized Submanifold
7．Totally Geodesic and Auto—Parallel Submanifolds
8．Normal Connectioand the WeingarteFormula
9．Vader Waerden—Bortolotti Connection
Chapter 4．Structural Equations．Local Symmetries
1．Torsioand Curvature Forms
2．CaftaStructural Equations iPolar Coordinates
3．Existence of Afne Local Mappings
4．Locally Symmetric Afne ConnectioSpaces
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．Reductioof Locally Symmetric Spaces to Globally Symmetric Spaces
6．Properties of Symmetries iGlobally Symmetric Spaces
7．Symmetric Spaces
8．Examples of Symmetric Spaces
9．Coincidence of Classes of Symmetric and Globally Symmetric Spaces
Chapter 6．Connections oLie Groups
1．Invariant Constructioof the Canonical Connection
2．Morphisms of Symmetric Spaces as Affine Mappings
3．Left—Invariant Connections oa Lie Group
4．CartaConnections
5．Left CartaConnection
6．Right—Invariant Vector Fields
7．Right CartaConnection
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．DynkiPolynomials
7．Local Lie Groups
8．Bijectivity of the Lie Functor
Chapter 8．Affine Fields and Related Topics
1．Affine Fields
2．Dimensioof the Lie Algebra of Affine Fields
3．Completeness of Affine Fields
4．Mappings of Left and Right Translatiooa Symmetric Space
5．Derivations oManifolds with Multiplication
6．Lie Algebra of Derivations
7．Involutive Automorphism of the DerivatioAlgebra of a Symmetric Space
8．Symmetric Algebras and Lie Ternaries
9．Lie Ternary of a Symmetric Space
Chapter 9．CartaTheorem
1．Functor s
2．Comparisoof the Functor s with the Lie Functor
3．Properties of the Functor s
4．Computatioof the Lie Ternary of the Space
5．Fundamental Group of the Quotient Space
6．Symmetric Space with a GiveLie Ternary
7．Coverings
8．CartaTheorem
9．Identificatioof Homogeneous Spaces with Quotient Spaces
10．Trauslations of a Symmetric Space
11．Proof of the CartaTheorem
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．TranslatioGroup of a Symmetric Space
Chapter 11．Lagrangians iRiemanniaSpaces
1．Riemanniaand Pseudo—RiemanniaSpaces
2．RiemanniaConnections
3．Geodesics ia RiemanniaSpace
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 ia RiemanniaSpace
2．Natural Parameter
3．RiemanniaDistance and Shortest Arcs
4．Extremals of the Length Lagrangian
5．RiemanniaCoordinates
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．RiemanniaVolume Element
2．Discriminant Tensor
3．Foss—Weyl Formula
4．Case n=2
5．Laplace Operator oa RiemanniaSpace
6．The GreeFormulas
7．Existence of Harmonic Functions with a Nonzero Differential
8．Conjugate Harmonic Functions
9．Isothermal Coordinates
10．Semi—CartesiaCoordinates
11．CartesiaCoordinates
……
Chapter 14．Minimal Surfaces
Chapter 15．Curvature iRiemanniaSpace
Chapter 16．GaussiaCurvature
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 oa Lie Group Ⅰ
Chapter 26．Metrics oa 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 oVector Bundles
Chapter 36．Curvature Tensor
Suggested Reading
Index
