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Lightlike Submanifolds Of Semi-Riemannian Manifolds

  • Krishan L. Duggal
  • Aurel Bejancu
Part of the Mathematics and Its Applications book series (MAIA, volume 364)

Abstract

The primary difference between the theory of lightlike submanifolds and the classical theory of Riemannian or semi-Riemannian submanifolds arises due to the fact that in the first case, a part of the normal vector bundle TM lies in the tangent bundle TM of the submanifold M of \(\bar M\), where as in the second case TMTM = {0}. Thus, first basic problem of the lightlike submaniflds is to replace the intersecting part by a vector subbundle whose sections are no where tangent to M. Following the technique used in Chapter 4, we use a screen distribution S(TM) on M and a screen vector bundle S(TM ) over M, to construct a transversal vector bundle tr(TM) of M. The general theory is much more involved and rather difficult than the special case of lightlike hypersurfaces discussed in Chapter 4, in particular reference to the dependence of all the induced geometric objects on the triplet (S(TM), S(TM ), tr(TM)). We obtain the structure equations of M which relate the curvature tensor of \(\bar M\) with the curvature tensors of the linear connections induced on the vector bundles involved in the study. Finally, we present some results on differential geometry of lightlike surfaces of Lorentz manifolds, in particular attention to the case when \(\bar M = \mathbb{R}_1^4\).

Keywords

Vector Bundle Fundamental Form Linear Connection Lorentz Manifold Coordinate Neighbourhood 
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 Dordrecht 1996

Authors and Affiliations

  • Krishan L. Duggal
    • 1
  • Aurel Bejancu
    • 2
  1. 1.University of WindsorWindsorCanada
  2. 2.Polytechnic Institute of IaşiIaşiRomania

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