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《几何6:黎曼几何》是一部值得一读的研究生教材，内容主要涉及黎曼几何基本定理的研究，如霍奇定理、Rauch比较定理、Lyusternik和Fet定理调和映射的存在性等，书中还有当代数学研究领域中的热门论题，有些内容则是首次出现在教科书中。
作者简介
作者：（俄罗斯）波斯特尼科夫（Postnikov M.M.）
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 （r， 1）
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
文摘
版权页：
插图：
《几何6:黎曼几何》是一部值得一读的研究生教材，内容主要涉及黎曼几何基本定理的研究，如霍奇定理、Rauch比较定理、Lyusternik和Fet定理调和映射的存在性等，书中还有当代数学研究领域中的热门论题，有些内容则是首次出现在教科书中。
作者简介
作者：（俄罗斯）波斯特尼科夫（Postnikov M.M.）
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 （r， 1）
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
文摘
版权页：
插图：
| ISBN | 9787030235039 |
|---|---|
| 出版社 | 科学出版社 |
| 作者 | 波斯特尼科夫 (Postnikov M.M.) |
| 尺寸 | 5 |