Reduction of the co dimension of Lightlike Isotropic Submanifolds.

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 et al. [Mathematics Lectures Series, Vol. 13, 1990], due to the degenerate metric on M. This result extends in some sense, the one in [J. Diff. Geom. 5 (1971) 333; Topology 25 (4) (1986) 541; Mathematics Lectures Series, Vol. 13, 1990] to lightlike isotropic submanifolds.