tailieunhanh - Relative nullity foliations and lightlike hypersurfaces in indefinite Kenmotsu manifolds

This paper deals with the relative nullity distributions of lightlike hypersurfaces of indefinite Kenmotsu space forms, tangent to the structure vector field. Theorems on parallel vector fields are obtained. We give characterization theorems for the relative nullity distributions as well as for Einstein, totally contact umbilical and flat lightlike hypersurfaces. | Turk J Math 35 (2011) , 129 – 149. ¨ ITAK ˙ c TUB doi: Relative nullity foliations and lightlike hypersurfaces in indefinite Kenmotsu manifolds Fortun´e Massamba Abstract This paper deals with the relative nullity distributions of lightlike hypersurfaces of indefinite Kenmotsu space forms, tangent to the structure vector field. Theorems on parallel vector fields are obtained. We give characterization theorems for the relative nullity distributions as well as for Einstein, totally contact umbilical and flat lightlike hypersurfaces. We show that, under a certain condition, Einstein lightlike hypersurfaces in indefinite Kenmotsu space forms have parallel screen distributions. We prove that on a parallel (or totally umbilical) lightlike hypersurface, the relative nullity space coincides with the tangent vector space. Key Words: η -Einstein lightlike hypersurfaces; Indefinite Kenmotsu manifold; Relative nullity foliation; Screen distribution. 1. Introduction Nullity spaces in Riemannian geometry have been studied by many authors, see references [1], [5] and references therein. Abe and Magid in [1], for instance, extended the study of the relative nullity to isometric immersion between manifolds with indefinite metric. The present paper aims to investigate a similar concept, namely, relative nullity foliations of lightlike hypersurfaces of indefinite Kenmotsu manifolds. Many differences from the Riemannian case are due to the fact that the metric in consideration is degenerate. Further advancements in this topic are recent (see [3], for instance). As is well known, contrary to timelike and spacelike hypersurfaces, the geometry of a lightlike hypersurface is different and rather difficult since the normal bundle and the tangent bundle have non-zero intersection. To overcome this difficulty, a theory on the differential geometry of lightlike hypersurfaces developed by Duggal and Bejancu [6] introduces a non-degenerate screen distribution and constructs the .