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

Manifold Glean 

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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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