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

Front Cover
Springer Science & Business Media, Aug 5, 1985 - Mathematics - 432 pages
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.
 

Contents

CHAPTER
1
2 The simplest examples of manifolds
10
3 Essential facts from the theory of Lie groups
20
4 Complex manifolds
31
5 The simplest homogeneous spaces
41
CHAPTER 2
65
10 Various properties of smooth maps of manifolds
77
11 Applications of Sards theorem
90
22 Covering homotopies The homotopy groups of covering spaces
193
23 Facts concerning the homotopy groups of spheres Framed normal
207
Smooth Fibre Bundles
220
25 The differential geometry of fibre bundles
251
26 Knots and links Braids
286
CHAPTER 7
297
28 Hamiltonian systems on manifolds Liouvilles theorem Examples
308
29 Foliations
322

CHAPTER 3
99
14 Applications of the degree of a mapping
110
15 The intersection index and applications
125
CHAPTER 4
135
18 Covering maps and covering homotopies
148
19 Covering maps and the fundamental group Computation of
157
CHAPTER 5
185
30 Variational problems involving higher derivatives
333
CHAPTER 8
358
32 Some examples of global solutions of the YangMills equations
393
33 The minimality of complex submanifolds
414
Index
423
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