CHAPTER 1 Examples of Manifolds
1. The concept of a manifold
1.1. Definition of a manifold
1.2. Mappings of manifolds; tensors on manifolds
1.3. Embeddings and immersions of manifolds. Manifolds with boundary
2. The simplest examples of manifolds
2.1. Surfaces in Euclidean space. Transformation groups as manifolds
2.2. Projective spaces
2.3. Exercises
3. Essential facts from the theory of Lie groups
3.1. The structure of a neighbourhood of the identity of a Lie group.The Lie algebra of a Lie group. Semisimplicity
3.2. The concept of a linear representation. An example of a non-matrix Lie group
4. Complex manifolds
4.1. Definitions and examples
4.2. Riemann surfaces as manifolds
5. The simplest homogeneous spaces
5.1. Action of a group on a manifold
5.2. Examples of homogeneous spaces
5.3. Exercises
6. Spaces of constant curvature (symmetric spaces)
6.1. The concept of a symmetric space
6.2. The isometry group of a manifold. Properties of its Lie algebra
6.3. Symmetric spaces of the first and second types
6.4. Lie groups as symmetric spaces
6.5. Constructing symmetric spaces. Examples
6.6. Exercises
7. Vector bundles on a manifold
7.1. Constructions involving tangent vectors
7.2. The normal vector bundle on a submanifold
CHAPTER 2 Foundational Questions. Essential Facts Concerning Functions on a Manifold. Typical Smooth Mappings
8. Partitions of unity and their applications
8.1. Partitions of unity
8.2. The simplest applications of partitions of unity. Integrals over a manifold and the general Stokes formula
8.3. Invariant metrics
9. The realization of compact manifolds as surfaces in RN
10. Various properties of smooth maps of manifolds
10.1. Approximation of continuous mappings by smooth ones
10.2. Sard's theorem
10.3. Transversal regularity
10.4. Morse functions
11. Applications of Sard's theorem
11.1. The existence of embeddings and immersions
11.2. The construction of Morse functions as height functions
11.3. Focal points
CHAPTER 3 The Degree of a Mapping. The Intersection Index of Submanifolds.Applications
12. The concept of homotopy
12.1. Definition of homotopy. Approximation of continuous maps and homotopies by smooth ones
12.2. Relative homotopies
13. The degree of a map
13.1. Definition of degree
13.2. Generalizations of the concept of degree
13.3. Classification of homotopy classes of maps from an arbitrary manifold to a sphere
13.4. The simplest examples
14. Applications of the degree of a mapping
14.1. The relationship between degree and integral
14.2. The degree of a vector field on a hypersurface
14.3. The Whitney number. The Gauss-Bonnet formula
14.4. The index of a singular point of a vector field
14.5. Transverse surfaces of a vector field. The Poincar6-Bendixson theorem
15. The intersection index and applications
15.1. Definition of the intersection index
15.2. The total index of a vector field
……
CHAPTER 4 Orientability of Manifolds. The Fundamental Group.Covering Spaces (Fibre Bundles with Discrete Fibre)
CHAPTER 5 Homotopy Groups
CHAPTER 6 Smooth Fibre Bundles
CHAPTER 7 Some Examples of Dynamical Systems and Foliations on Manifolds
CHAPTER 8 The Global Structure of Solutions of Higher-Dimensional Variational Problems
Bibliography
Index
