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《几何1:微分几何基本思想与概念》由科学出版社出版。
作者简介
作者：（俄罗斯）加姆克列利泽（R.V.Gamkrelidze）
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
文摘
版权页：
插图：
《几何1:微分几何基本思想与概念》由科学出版社出版。
作者简介
作者：（俄罗斯）加姆克列利泽（R.V.Gamkrelidze）
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
文摘
版权页：
插图：
| ISBN | 9787030234988 |
|---|---|
| 出版社 | 科学出版社 |
| 作者 | 加姆克列利泽 (R.V.Gamkrelidze) |
| 尺寸 | 5 |