Algebraicity of Induced Riemannian Curvature Tensor on Lightlike Warped Product Manifolds

Abstract

Lightlike warped product manifolds are considered in this paper. The geometry of lightlike submanifolds is difficult to study since the normal vector bundle intersects with the tangent bundle. Due to the degenerate metric, the induced connection is not metric and it follows that the Riemannian curvature tensor is not algebraic. In this situation, some basic techniques of calulus are not useable. In this paper, we consider lightlike warped product as submanifold of semi-Riemannian manifold and establish some remarkable geometric properties from which we establish some conditions on the algebraicity of the induced Riemannian curvature tensor.

Osserman Conditions in Lightlike Warped Product Geometry

“…, it is obvious that N is a conformally Osserman manifold. The lightlike warped product metric g belongs to the conformal class of Due to Proposition 2 in [12], we establish the following two results for coisotropic warped product ( )…”

“…Moreover 1 N is conformally Osserman. By Theorem 5 in [12], R is an algebraic curvature tensor. If we restrict our study on the product ( )…”

“…From Proposition 2, theorem 5 in [12] and Theorem 4.3 in [9], we proved the following result that characterizes any screen distribution of a coisotropic warped product of a semi-Riemannian space form with the first factor totally null. This case consists of a class of null warped products that is Einstein and pointwise Osserman.…”

“…It is obvious that M is a coisotropic warped product submanifold of 6 3  with warping function 1 v ρ = . Using (12) and (17) we obtain…”

Inequalities for Scalar Curvature and Shape Operator of an R-Lightlike Submanifold in Semi-Riemannian Manifold

“…In [7] the autors established some remarkable geometric roperties to ensure algebraicity of the induced Riemannian curvature tensor on lightlike Warped Product Manifolds. The same approach has been explored in [8] to present osserman conditions for lightlike warped product manifolds.…”