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 |