Geometry of lightlike submanifolds. Lightlike hypersurfaces and isotropic submanifolds,

Abstract

In a recent past, the growing importance of lightlike submanifolds in global Lorentzian geometry and their extensive use in general relativity, motivated their study in a semi-Riemannian manifold. This is the lightlike geometry of (sub-)manifolds where there are significant differences with the nondegenerate case and who make its study slightly more complicated. Indeed, one faces significant technical challenges in their study because conventional techniques known in the nondegenerate case fail. As a consequence, while the geometry of nondegenerate (semi-)Riemannian (sub-)manifolds is almost entirely developed and is well understood, its degenerate counterpart is relatively new and not well explored. So considerable works are needed to fill the gap. The present book falls into this category. It introduces a basic concept: the pseudo-inversion of degenerate metrics which turns out to be decisive whenever the inversion of the metric is required, and we carry out interesting applications. Screen conformal normalization along with Einstein condition are studied. For lightlike isotropic submanifolds, we consider the problem of reduction of codimension.