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Line fields and Lorentz manifolds

  • W. H. Greub
4. Geometric Methods and Global Analysis
Part of the Lecture Notes in Physics book series (LNP, volume 139)

Abstract

The first result of this paper is that the total Gaussian curvature of a compact Lorentz manifold vanishes (Theorem I, sec. 8). The argument in the proof is different from those given in [1] and [2]. In fact, we make extensive use of the existence of a global line field on a Lorentz manifold, rather than reducing the problem to the classical case of a Riemannian manifold.

In §4 the index of a line field at an isolated singularity is classical case of a Riemannian manifold. sum of a line field on a compact Riemannian 4-manifold in terms of the total Gaussian curvature. A consequence of this result is that every compact 4-manifold which admits a line field, also admits a vector field without zeros. (Theorem III, sec. 13). Finally, in sec. 15 it is shown that the notions of orientability and time-orientability on a Lorentz manifold are independent.

Keywords

Vector Field Riemannian Manifold Vector Bundle Line Field Fibre Bundle 
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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Bibliography

  1. [1]
    Avez, A., Formule de Gauss-Bonnet-Chern en métrigue de signature quelquongeu, C. R. Acad, Sci., Paris 255 (1962)Google Scholar
  2. [2]
    Chern, S.S., Pseudo-Riemannian geometry and Gauss-Bonnet formula, An. Acad. Brazil Ci. 35 (1963).Google Scholar
  3. [3]
    Greub, W., Halperin, S. and Vanstone, R., Connections, Curvature and cohomology, volume I, Academic Press, New York, 1972.Google Scholar
  4. [4]
    Greub, W., Halperin, S. and Vanstone, R., Connections, Curvature and Cohomology, volume II, Academic Press, New York, 1973.Google Scholar
  5. [5]
    Marcus, L., Line element fields and Lorentz structures on differntiable manifolds, Ann. of Math. 8, volume 83, 1956.Google Scholar
  6. [6]
    Samelson, H., A theorem on differentiable manifolds, Portigaliae Math., volume 10 (1951).Google Scholar

Copyright information

© Springer-Verlag 1981

Authors and Affiliations

  • W. H. Greub
    • 1
  1. 1.University of TorontoToronto

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