C. Atindogbe, J.-P. Ezin, and Joël Tossa

Copyright © 2003 C. Atindogbe et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

Let (M,g) be a smooth manifold M endowed with a metric g. A large class of differential operators in differential geometry is intrinsically defined by means of the dual metric g∗ on the dual bundle TM∗ of 1-forms on M. If the metric g is (semi)-Riemannian, the metric g∗ is just the inverse of g. This paper studies the definition of the above-mentioned geometric differential operators in the case of manifolds endowed with degenerate metrics for which g∗ is not defined. We apply the theoretical results to Laplacian-type operator on a lightlike hypersurface to deduce a Takahashi-like theorem (Takahashi (1966)) for lightlike hypersurfaces in Lorentzian space ℝ1n+2.