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.

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.