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
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