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 Higher-Dimensional Variational Problems. - 31. Some manifolds arising in the general theory of relativity (GTR). - 32. Some examples of global solutions of the Yang-Mills equations. Chiral fields. - 33. The minimality of complex submanifolds. Language: English