a-Associated metrics on rigged null hypersurfaces

Abstract

Let x : (M; g) ! (M; g) be a null hypersurface isometrically immersed into a proper semi-Riemannianmanifold (M; g) . A rigging for M is a vector field defined on some open subset of M containing M such thatp /2TpM for every p 2 M. Such a vector field induces an everywhere transversal null vector field N defined over Mand which induces on M the same geometrical objects as . Let be the 1-form that is g -metrically equivalent to Nand let = x⋆ be its pullback on M. For a given nowhere vanishing smooth function on M, we have introduced andstudied the so-called -associated (semi-)Riemannian metric g = g+ . It turns out that this perturbation of theinduced metric along a transversal null vector field is always nondegenerate, so we have established some relationshipsbetween geometrical objects of the (semi-)Riemannian manifold (M; g) and those of the lightlike hypersurface (M; g) .For instance, in the case where N is closed, we give a constructive method to find a -associated metric whose Levi-Civita connection coincides with the connection ∇ induced on M through the projection of the Levi-Civita connection∇ of M along N . As an application, we show that given a null Monge hypersurface M in Rn+1q ; there always exists arigging and a -associated metric whose Levi-Civita connection is the induced connection on M.