Abstract
This paper deals with a family of Osserman lightlike hypersurfaces $(M_u)$ of a class of Lorentzian manifolds $\bar{M}$ such that its each null normal vector is defined on some open subset of $\bar{M}$ around $M_u$. We prove that a totally umbilical family of lightlike hypersurfaces of a connected Lorentzian pointwise Osserman manifold of constant curvature is locally Einstein and pointwise ${\cal F}-$Osserman, where our foliation approach provides the required algebraic symmetries of the induced curvature tensor. Also we prove two new characterization theorems for the family of Osserman lightlike hypersurfaces, supported by a physical example of Osserman lightlike hypersurfaces of the Schwarzschild spacetime.
Highlights
A primary interest in differential geometry is to determine the curvature and the metric of a given smooth manifold, which distinguishes the geometry of this subject from the others that are analytic, algebraic or topological
This paper deals with a family of Osserman lightlike hypersurfaces (Mu) of a class of Lorentzian manifolds Msuch that its each null normal vector is defined on some open subset of Maround Monge hypersurfaces (Mu)
Is there a better way to deal with the lightlike Osserman geometry to improve on previous works on this topic and find some new results? In this paper we answer both of these questions by using the following different approach: Consider a class of lightlike hypersurfaces M for which we assume that the null normal ξ is defined on some open subset of Maround M
Summary
A primary interest in differential geometry is to determine the curvature and the metric of a given smooth manifold, which distinguishes the geometry of this subject from the others that are analytic, algebraic or topological. Since in Duggal-Bejancu approach the lightlike geometry depends on a choice of screen distribution S (T M) which is not unique, a well-defined concept of Osserman condition is not possible for an arbitrary lightlike hypersurface They looked for an admissible S (T M) for which the associated induced curvature tensor of M is an algebraic curvature. In this paper we answer both of these questions by using the following different approach: Consider a class of lightlike hypersurfaces M for which we assume that the null normal ξ is defined on some open subset of Maround M. It follows that N as given in (2.6) is a null transversal normalizing vector field of M satisfying the second equality in (2.7)
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