zbMATH — the first resource for mathematics

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.

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