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
We consider a class of lightlike hypersurfaces M, of semi-Riemannian manifolds, whose shape operator is conformal to the shape operator of its screen distributions. Integrability and characterization results are then established.
We study Weyl structures on lightlike hypersurfaces endowed with a conformal structure of certain type and specific screen distribution: the Weyl screen structures. We investigate various differential geometric properties of Einstein-Weyl screen structures on lightlike hypersurfaces and show that, for ambient Lorentzian space R +2 1 and a totally umbilical screen foliation, there is a strong interplay with the induced (Riemannian) Weyl-structure on the leaves.
MSC:53C50, 53C05, 53C25
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