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

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

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

Keywords:

Lie groups; complex manifolds; homogeneous spaces; symmetric spaces; space forms; Sard’s theorem; surfaces; Morse functions; Gauss-Bonnet formula; Poincaré Bendixson theory; index of vector field; Lefschetz fixed point index; Hopf invariant; Whitehead product; fibrations; suspensions; homotopy groups of spheres; characteristic classes; knots; links; braids; connections; curvature; Hamiltonian systems; geodesic flows; Liouville’s theorem; Korteweg-deVries equation; cosmological models; Yang-Mills equations; chiral fields; functions on a manifold; smooth mappings; degree of a mapping; intersection index; Orientability; fundamental group; Covering spaces; Homotopy groups; Smooth fibre bundles; dynamical systems; foliations; higher-dimensional variational problems