International Journal of Mathematics and Mathematical Sciences
Volume 2003 (2003), Issue 55, Pages 3479-3501
doi:10.1155/S0161171203301309

Pseudoinversion of degenerate metrics

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.