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Modern Geometry

Modern Geometry

Modern Geometry

Modern Geometry

Modern Geometry

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Author B.A.Dubrovin, A.T.Fomenko, S.P.Novikov

Publisher World Publishing Corporation

Isbn

750620133x, 9787506201339

Published Nov 1999

Pages 430

Size 24k

Edition first

Format Paperback

Language English

Page material copper plate paper

Product Description:

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 levelof abstractness of their exposition

Content:

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 neighbourbood 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 RN

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

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. 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 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

23.2. The suspension map

23.3. Calculation of the groups

23.4. The groups

Chapter 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

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

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

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 submanifotds

Bibliography

Index

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