Modern Geometry— Methods and Applications Part II: The Geometry and Topology of Manifolds

Up until recently, Riemannian geometry and basic topology were not included, even by departments or faculties of mathematics, as compulsory subjects in a university-level mathematical education. The standard courses in the classical differential geometry of curves and surfaces which were given instead (and still are given in some places) have come gradually to be viewed as anachronisms. However, there has been hitherto no unanimous agreement as to exactly how such courses should be brought up to date, that is to say, which parts of modern geometry should be regarded as absolutely essential to a modern mathematical education, and what might be the appropriate level of abstractness of their exposition. The task of designing a modernized course in geometry was begun in 1971 in the mechanics division of the Faculty of Mechanics and Mathematics of Moscow State University. The subject-matter and level of abstractness of its exposition were dictated by the view that, in addition to the geometry of curves and surfaces, the following topics are certainly useful in the various areas of application of mathematics (especially in elasticity and relativity, to name but two), and are therefore essential: the theory of tensors (including covariant differentiation of them); Riemannian curvature; geodesics and the calculus of variations (including the conservation laws and Hamiltonian formalism); the particular case of skew-symmetric tensors (i. e.

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.