Alpha-Associated metrics on rigged null hypersurfaces

Abstract

Let $x:(M,g)\to(\Bm,\bg)$ be a null hypersurface isometrically immersed into a proper semi-Riemannian manifold $(\overline{M},\bar g)$. A rigging for $M$ is a vector field $\zeta$ defined on some open subset of $\overline{M}$ containing $M$ such that $\zeta_p\notin T_pM$ for every $p\in M$. Such a vector field induces an everywhere transversal null vector field $N$ defined over $M$ and which induces on $M$ the same geometrical objects as $\zeta$. Let $\bar \eta$ be the 1-form that is $\bar g$-metrically equivalent to $N$ and let $\eta=x^\star\bar\eta$ be its pullback on $M$. For a given nowhere vanishing smooth function $\alpha$ on $M$, we have introduced and studied the so-called $\alpha$-associated (semi-)Riemannian metric $g_{\alpha}=g+\alpha\eta\otimes \eta$. It turns out that this perturbation of the induced metric along a transversal null vector field is always nondegenerate, so we have established some relationships between geometrical objects of the (semi-)Riemannian manifold $(M,{g}_{\alpha})$ 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 $\alpha$-associated metric whose Levi-Civita connection coincides with the connection $\nabla$ induced on $M$ through the projection of the Levi-Civita connection $\overline{\nabla}$ of $\overline{M}$ along $N$. As an application, we show that given a null Monge hypersurface $M$ in $\R_q^{n+1},$ there always exists a rigging and a $\alpha$-associated metric whose Levi-Civita connection is the induced connection on $M$.