Global lightlike manifolds and harmonicity

Abstract

A new class of semi-Riemannian and lightlike manifolds (including globally null) is constructed by using a hypersurface of an orientable Riemannian manifold, endowed with the second fundamental form instead of a metric induced from the ambient space. We show the existence (or non-existence) of harmonic tensor fields and harmonic maps and extend to the semi-Riemannian and lightlike case a result of Chen-Nagano [4]. Then we deal with general lightlike submanifolds immersed in a semi-Riemannian manifold and propose a definition of minimal lightlike submanifolds, which generalize the one given in [7] in the Minkowski space ${\bf R}^4_1$. Several examples are given throughout.