Pseudo-inversion 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 T M ∗ 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 R^n+2_{1} .