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

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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.

Author(s): B. A. Dubrovin, S. P. Novikov, A. T. Fomenko (auth.)
Series: Graduate Texts in Mathematics 104
Edition: 1
Publisher: Springer-Verlag New York
Year: 1985

Language: English
Pages: 432
Tags: Manifolds and Cell Complexes (incl. Diff.Topology); Differential Geometry

Front Matter....Pages i-xv
Examples of Manifolds....Pages 1-64
Foundational Questions. Essential Facts Concerning Functions on a Manifold. Typical Smooth Mappings....Pages 65-98
The Degree of a Mapping. The Intersection Index of Submanifolds. Applications....Pages 99-134
Orientability of Manifolds. The Fundamental Group. Covering Spaces (Fibre Bundles with Discrete Fibre)....Pages 135-184
Homotopy Groups....Pages 185-219
Smooth Fibre Bundles....Pages 220-296
Some Examples of Dynamical Systems and Foliations on Manifolds....Pages 297-357
The Global Structure of Solutions of Higher-Dimensional Variational Problems....Pages 358-418
Back Matter....Pages 419-432