Modern Geometry - Methods and Applications

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.- 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 ?N.- §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 86 §.- 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.- 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 Poincaré-Bendixson theorem.- §15. The intersection index and applications.- 15.1. Definition of the intersection index.- 15.2. The total index of a vector field.- 15.3. The signed number of fixed points of a self-map (the Lefschetz number). The Brouwer fixed-point theorem.- 15.4. The linking coefficient.- 4 Orientability of Manifolds. The Fundamental Group. Covering Spaces (Fibre Bundles with Discrete Fibre).- §16. Orientability and homotopies of closed paths.- 16.1. Transporting an orientation along a path.- 16.2. Examples of non-orientable manifolds.- §17. The fundamental group.- 17.1. Definition of the fundamental group.- 17.2. The dependence on the base point.- 17.3. Free homotopy classes of maps of the circle.- 17.4. Homotopic equivalence.- 17.5. Examples.- 17.6. The fundamental group and orientability.- §18. Covering maps and covering homotopies.- 18.1. The definition and basic properties of covering spaces.- 18.2. The simplest examples. The universal covering.- 18.3. Branched coverings. Riemann surfaces.- 18.4. Covering maps and discrete groups of transformations.- §19. Covering maps and the fundamental group. Computation of the fu