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Orientability of Manifolds. The Fundamental Group. Covering Spaces (Fibre Bundles with Discrete Fibre)

  • B. A. Dubrovin
  • S. P. Novikov
  • A. T. Fomenko
Part of the Graduate Texts in Mathematics book series (GTM, volume 104)

Abstract

According to the simplest of the definitions of an orientation on a manifold given above (see Definition 1.1.3), a manifold M is oriented if the local coordinate systems x α j given on the members U j of a covering collection of local co-ordinate neighbourhoods (or charts) for M, are such that the transition functions from one local co-ordinate system to another on the regions of overlap U j U k , have positive Jacobian:
$$\det \left( {\frac{{\partial x_j^\alpha }}{{\partial x_k^\beta }}} \right) > 0.$$

Keywords

Riemann Surface Fundamental Group Discrete Group Homotopy Class Fuchsian Group 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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Copyright information

© Springer Science+Business Media New York 1985

Authors and Affiliations

  • B. A. Dubrovin
    • 1
  • S. P. Novikov
    • 2
  • A. T. Fomenko
    • 3
  1. 1.Department of Mathematics and MechanicsMoscow UniversityMoscowRussia
  2. 2.Institute of Physical Sciences and TechnologyMaryland UniversityCollege ParkUSA
  3. 3.Moscow State UniversityMoscowRussia

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