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Modern geometry - methods and applications. Part 2: The geometry and topology of manifolds. Transl. from the Russian by Robert G. Burns. (English) Zbl 0565.57001
Graduate Texts in Mathematics, 104. New York etc.: Springer-Verlag. XV, 430 p. DM 158.00 (1985).
This is a translation of the second part of the Russian original (1979; Zbl 0433.53001). It contains the following chapters: Examples of manifolds; Foundational questions. Essential facts concerning functions on a manifold. Typical smooth mappings; The degree of a mapping. The intersection index of submanifolds. Applications; Orientability of manifolds. The fundamental group. Covering spaces; Homotopy groups; Smooth fibre bundles; Some examples of dynamical systems and foliations on manifolds; The global structure of solutions of higher-dimensional variational problems.

##### MSC:
 57-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to manifolds and cell complexes 55-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to algebraic topology 53-01 Introductory exposition (textbooks, tutorial papers, etc.) pertaining to differential geometry 70G10 Generalized coordinates; event, impulse-energy, configuration, state, or phase space for problems in mechanics 70H05 Hamilton’s equations 70Sxx Classical field theories 81T08 Constructive quantum field theory 83F05 Cosmology 22E10 General properties and structure of complex Lie groups 22E15 General properties and structure of real Lie groups 32Q99 Complex manifolds 53C05 Connections, general theory 53C20 Global Riemannian geometry, including pinching 53C22 Geodesics in global differential geometry 53C30 Differential geometry of homogeneous manifolds 53C35 Differential geometry of symmetric spaces 53A05 Surfaces in Euclidean and related spaces 55Q05 Homotopy groups, general; sets of homotopy classes 55Q15 Whitehead products and generalizations 55Q25 Hopf invariants 55M20 Fixed points and coincidences in algebraic topology 55M25 Degree, winding number 55Q40 Homotopy groups of spheres 55R05 Fiber spaces in algebraic topology 55R10 Fiber bundles in algebraic topology 55R25 Sphere bundles and vector bundles in algebraic topology 55R40 Homology of classifying spaces and characteristic classes in algebraic topology 57M05 Fundamental group, presentations, free differential calculus 57M10 Covering spaces and low-dimensional topology 57M25 Knots and links in the $$3$$-sphere (MSC2010) 57N80 Stratifications in topological manifolds 57R20 Characteristic classes and numbers in differential topology 57R30 Foliations in differential topology; geometric theory 57R35 Differentiable mappings in differential topology 57R70 Critical points and critical submanifolds in differential topology 57R25 Vector fields, frame fields in differential topology 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems 37J35 Completely integrable finite-dimensional Hamiltonian systems, integration methods, integrability tests 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 37D40 Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.) 53D25 Geodesic flows in symplectic geometry and contact geometry 58J35 Heat and other parabolic equation methods for PDEs on manifolds 58J45 Hyperbolic equations on manifolds 58E30 Variational principles in infinite-dimensional spaces