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.