Curvature properties of some class of warped product manifolds

TL;DR

This paper investigates the curvature properties of certain warped product manifolds, demonstrating specific proportionality relations and curvature conditions, especially for 2-dimensional bases with constant curvature fibers, including applications to non-Einstein spacetimes.

Contribution

It establishes new curvature proportionality relations for warped products with 2-dimensional bases and constant curvature fibers, expanding understanding of pseudosymmetry conditions in these manifolds.

Findings

Curvature tensors are proportional to specific Kulkarni-Nomizu products.

Non-conformally flat, non-Einstein warped spacetimes satisfy these curvature conditions.

Curvature properties of quasi-Einstein manifolds and the Goedel spacetime are characterized.

Abstract

Warped product manifolds with p-dimensional base, p=1,2, satisfy some curvature conditions of pseudosymmetry type. These conditions are formed from the metric tensor g, the Riemann-Christoffel curvature tensor R, the Ricci tensor S and the Weyl conformal curvature C of the considered manifolds. The main result of the paper states that if p=2 and the fibre is a semi-Riemannian space of constant curvature, if n is greater or equal to 4, then the (0,6)-tensors R.R - Q(S,R) and C.C of such warped products are proportional to the (0,6)-tensor Q(g,C) and the tensor C is expressed by a linear combination of some Kulkarni-Nomizu products formed from the tensors g and S. Thus these curvature conditions satisfy non-conformally flat non-Einstein warped product spacetimes (p=2, n=4). We also investigate curvature properties of pseudosymmetry type of quasi-Einstein manifolds. In particular, we obtain some curvature property of the Goedel spacetime.