Reduction of the co-dimension of light-like isotropic sub-manifolds

Abstract

We give a sufficient condition for a lightlike isotropic submanifold $M$, of dimension $n$, which is not totally geodesic in a semi-Riemannian manifold of constant curvature $c$ and of dimension $n+p (n < p)$, to admit a reduction of codimension. We show that this condition is a necessary and sufficient condition on the first transversal space of $M$. There are basic and non-trivial differences from the Riemannian case, as developed by Dajczer \textit{et al} in (\cite{dajczer}), due to the degenerate metric on $M$. This result extends in some sense,the one in \cite{keti} and \cite{dajczer} to lightlike isotropic submanifolds.