A Short Course in Differential Geometry and Topology

Contents
Preface
1 Introduction to Differential Geometry
1.1 Curvilinear Coordinate Systems. Simplest Examples
1.1.1 Motivation
1.1.2 Cartesian and curvilinear coordinates
1.1.3 Simplest examples of curvilinear coordinates
1.2 The Length of a Curve in Curvilinear Coordinates
1.2.1 The length of a curve in Euclidean coordinates
1.2.2 The length of a curve in curvilinear coordinates
1.2.3 Concept of a Riemannian metric in a domain of Euclidean space
1.2.4 Indefinite metrics
1.3 Geometry on a Sphere and Plane
1.4 Pseudo-Sphere and Lobachevskii Geometry
2 General Topology
2.1 Definitions and Simplest Properties of Metric and Topological Spaces
2.1.1 Metric spaces
2.1.2 Topological spaces
2.1.3 Continuous mappings
2.1.4 Quotient topology
2.2 Connectedness. Separation Axioms
2.2.1 Connectedness
2.2.2 Separation Axioms
2.3 Compact Spaces
2.3.1 Compact spaces
2.3.2 Properties of compact spaces
2.3.3 Compact metric spaces
2.3.4 Operations on compact spaces
2.4 Functional Separability
2.4.1 Functional separability
2.4.2 Partition of unity
3 Smooth Manifolds (General Theory)
3.1 Concept of a Manifold
3.1.1 Main definitions
3.1.2 Functions of a change of coordinates. Definition of a smooth manifold
3.1.3 Smooth mappings. Diffeomorphism
3.2 Assignment of Manifolds by Equations
3.3 Tangent Vectors. Tangent Space
3.3.1 Simplest examples
3.3.2 General definition of tangent vector
3.3.3 Tangent space T_P_0 (M)
3.3.4 Directional derivative of a function
3.3.5 Tangent bundle
3.4 Submanifolds
3.4.1 Differential of a smooth mapping
3.4.2 Local properties of mappings and the differential
3.4.3 Embedding of manifolds in Euclidean space
3.4.4 Riemannian metric on a manifold
3.4.5 Sard theorem
4 Smooth Manifolds (Examples)
4.1 Theory of Curves on the Plane and in the Three-Dimensional Space
4.1.1 Theory of curves on the plane. Frénet Formulae
4.1.2 Theory of spatial curves. Frénet formulae
4.2 Surfaces. First and Second Quadratic Forms
4.2.1 First quadratic form
4.2.2 Second quadratic form
4.2.3 Elementary theory of smooth curves on a hypersurface
4.2.4 Gaussian and mean curvatures of two-dimensional surfaces
4.3 Transformation Groups
4.3.1 Simplest examples of transformation groups
4.3.2 Matrix transfromation groups
4.3.3 General linear group
4.3.4 Special linear group
4.3.5 Orthogonal group
4.3.6 Unitary group and special unitary group
4.3.7 Symplectic compact and noncompact groups
4.4 Dynamical Systems
4.5 Classification of Two-Dimensional Surfaces
4.5.1 Manifolds with boundary
4.5.2 Orientable manifolds
4.5.3 Classification of two-dimensional manifolds
4.6 Two-Dimensional Manifolds as Riemann Surfaces of Algebraic Functions
5 Tensor Analysis
5.1 General Concept of Tensor Field on a Manifold
5.2 Simplest Examples of Tensor Fields
5.2.1 Examples
5.2.2 Algebraic operations on tensors
5.2.3 Skew-symmetric tensors
5.3 Connection and Covariant Differentiation
5.3.1 Definition and properties of affine connection
5.3.2 Riemannian connection
5.4 Parallel Translation. Geodesics
5.4.1 Preparatory remarks
5.4.2 Equation of parallel translation
5.4.3 Geodesics
5.5 Curvature Tensor
5.5.1 Preparatory remarks
5.5.2 Coordinate definition of a curvature tensor
5.5.3 Invariant definition of a curvature tensor
5.5.4 Algebraic properties of the Riemannian curvature tensor
5.5.5 Some applications of the Riemannian curvature tensor
6 Homology Theory
6.1 Calculus of Differential Forms. Chomologies
6.1.1 Differential properties of exterior forms
6.1.2 Cohomologies of a smooth manifold (de Rham cohomologies)
6.2 Integration of Exterior Forms
6.2.1 Integral of a differential form over a manifold
6.2.2 Stokes formula
6.3 Degree of a Mapping and Its Applications
6.3.1 Degree of a mapping
6.3.2 Main theorem of algebra
6.3.3 Integration of forms
6.3.4 Gaussian mapping of a hypersurface
7 Simples Variational Problems of Riemannian Geometry
7.1 Concept of Functional. Extremal Functions. Eular Equation
7.2 Extremality of Geodesics
7.3 Minimal Surfaces
7.4 Calculus of Variations and Symplectic Geometry
Bibliography
Index