Modern Geometry¿ Methods and Applications

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.- 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 fundamental group of certain manifolds.- 19.1. Monodromy.- 19.2. Covering maps as an aid in the calculation of fundamental groups.- 19.3. The simplest of the homology groups.- 19.4. Exercises.- §20. The discrete groups of motions of the Lobachevskian plane.- 5 Homotopy Groups.- §21. Definition of the absolute and relative homotopy groups. Examples.- 21.1. Basic definitions.- 21.2. Relative homotopy groups. The exact sequence of a pair.- §22. Covering homotopies. The homotopy groups of covering spaces and loop spaces.- 22.1. The concept of a fibre space.- 22.2. The homotopy exact sequence of a fibre space.- 22.3. The dependence of the homotopy groups on the base point.- 22.4. The case of Lie groups.- 22.5. Whitehead multiplication.- §23. Facts concerning the homotopy groups of spheres. Framed normal bundles. The Hopf invariant.- 23.1. Framed normal bundles and the homotopy groups of spheres.- 23.2. The suspension map.- 23.3. Calculation of the groups ?n+1(Sn).- 23.4. The groups ?n+2(Sn).- 6 Smooth Fibre Bundles.- §24. The homotopy theory of fibre bundles.- 24.1. The concept of a smooth fibre bundle.- 24.2. Connexions.- 24.3. Computation of homotopy groups by means of fibre bundles.- 24.4. The classification of fibre bundles.- 24.5. Vector bundles and operations on them.- 24.6. Meromorphic functions.- 24.7. The Picard-Lefschetz formula.- §25. The differential geometry of fibre bundles.- 25.1. G-connexions on principal fibre bundles.- 25.2. G-connexions on associated fibre bundles. Examples.- 25.3. Curvature.- 25.4. Characteristic classes: Constructions.- 25.5. Characteristic classes: Enumeration.- §26. Knots and links. Braids.- 26.1. The group of a knot.- 26.2. The Alexander polynomial of a knot.- 26.3. The fibre bundle associated with a knot.- 26.4. Links.- 26.5. Braids.- 7 Some Examples of Dynamical Systems and Foliations on Manifolds.- §27. The simplest concepts of the qualitative theory of dynamical systems. Two-dimensional manifolds.- 27.1. Basic definitions.- 27.2. Dynamical systems on the torus.- §28. Hamiltonian systems on manifolds. Liouville’s theorem. Examples.- 28.1. Hamiltonian systems on cotangent bundles.- 28.2. Hamiltonian systems on symplectic manifolds. Examples.- 28.3. Geodesic flows.- 28.4. Liouville’s theorem.- 28.5. Examples.- §29. Foliations.- 29.1. Basic definitions.- 29.2. Examples of foliations of codimension 1.- §30. Variational problems involving higher derivatives.- 30.1. Hamiltonian formalism.- 30.2. Examples.- 30.3. Integration of the commutativity equations. The connexion with the Kovalevskaja problem. Finite-zoned periodic potentials.- 30.4. The Korteweg-deVries equation. Its interpretation as an infinite-dimensional Hamiltonian system.- 30.5 Hamiltonian formalism of field systems.- 8 The Global Structure of Solutions of Higher-Dimensional Variational Problems.- §31. Some manifolds arising in the general theory of relativity (GTR).- 31.1. Statement of the problem.- 31.2. Spherically symmetric solutions.- 31.3. Axially symmetric solutions.- 31.4. Cosmological models.- 31.5. Friedman’s models.- 31.6. Anisotropic vacuum models.- 31.7. More general models.- §32. Some examples of global solutions of the Yang-Mills equations. Chiral fields.- 32.1. General remarks. Solutions of monopole type.- 32.2. The duality equation.- 32.3. Chiral fields. The Dirichlet integral.- §33. The minimality of complex submanifolds.