Modern Geometry- Methods and Applications

1 Examples of Manifolds.- 1. The concept of a manifold.- 2. The simplest examples of manifolds.- 3. Essential facts from the theory of Lie groups.- 4. Complex manifolds.- 5. The simplest homogeneous spaces.- 6. Spaces of constant curvature (symmetric spaces).- 7. Vector bundles on a manifold.- 2 Foundational Questions. Essential Facts Concerning Functions on a Manifold. Typical Smooth Mappings.- 8. Partitions of unity and their applications.- 9. The realization of compact manifolds as surfaces in ?N.- 10. Various properties of smooth maps of manifolds.- 11. Applications of Sard's theorem.- 3 The Degree of a Mapping. The Intersection Index of Submanifolds. Applications.- 12. The concept of homotopy.- 13. The degree of a map.- 14. Applications of the degree of a mapping.- 15. The intersection index and applications.- 4 Orientability of Manifolds. The Fundamental Group. Covering Spaces (Fibre Bundles with Discrete Fibre).- 16. Orientability and homotopies of closed paths.- 17. The fundamental group.- 18. Covering maps and covering homotopies.- 19. Covering maps and the fundamental group. Computation of the fundamental group of certain manifolds.- 20. The discrete groups of motions of the Lobachevskian plane.- 5 Homotopy Groups.- 21. Definition of the absolute and relative homotopy groups. Examples.- 22. Covering homotopies. The homotopy groups of covering spaces and loop spaces.- 23. Facts concerning the homotopy groups of spheres. Framed normal bundles. The Hopf invariant.- 6 Smooth Fibre Bundles.- 24. The homotopy theory of fibre bundles.- 25. The differential geometry of fibre bundles.- 26. Knots and links. Braids.- 7 Some Examples of Dynamical Systems and Foliations on Manifolds.- 27. The simplest concepts of thequalitative theory of dynamical systems. Two-dimensional manifolds.- 28. Hamiltonian systems on manifolds. Liouville's theorem. Examples.- 29. Foliations.- 30. Variational problems involving higher derivatives.- 8 The Global Structure of Solutions of