Modern Geometry. Methods and Applications: Part 2: The Geometry and Topology of Manifolds
Graduate Texts in Mathematics 93

Sğringår | 1985 | ISBN: 0387961623 3540961623 9780387961620 | 445 pages | PDF/djvu | 6/5 MB
This is of an introduction to modern geometry, with emphasis on applications to other areas of mathematics and theoretical physics. Topics covered include tensors and their differential calculus, the calculus of variations in one and several dimensions, and geometric field theory. This material is explained in as simple and concrete a language as possible, in a terminology acceptable to physicists. The text for the second edition has been substantially revised.

Contents
CHAPTER 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
CHAPTER 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 lR/Y
§1O. 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
§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
CHAPTER 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. Oassification 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 Poincare-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
CHAPTER 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
CHAPTER 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
23.3. Calculation of the groups XIl+1(SIl)
23.4. The groups XIl +2(SIl)
CHAPTER 6 Smooth Fibre Bundles
§24. The homotopy theory of fi bre bundl
24.1. The concept of a smooth fibre bundle 220
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
CHAPTER 7 Some Examples of Dynamical Systems and Foliations on Manifolds
§27. The simplest concepts of the qualitative theory of dynamical systems. Two-dimensional manifolds 297
27.1. Basic definitions
27.2. Dynamical systems on the torus
§28. Hamiltonian systems on manifolds. LiouviHe's theorem. Examples
28.1. Hamiltonian systems on cotangent bundles
28.2. Hamiltonian systems on symplectic manifolds. ExampJes
28.3. Geodesic ftows
28.4. LiouviUe'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
CHAPTER 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
Bibliography
Index