Solutions of Einstein’s equations with conformal symmetries and a contact slice

Abstract

First we show that the initial space-like slice of the generalized Robertson–Walker (GRW) space–time is Einstein if and only if the electric part of the space–time Weyl conformal tensor vanishes. We also show that the purely spatial component of the space–time Weyl tensor vanishes if and only if the initial slice has constant curvature. Then, assuming that a spatially complete synchronous space–time solution of Einstein’s equations admits a conformal vector field whose normal component is non-constant and spatial component is conformal on each space-like slice, we show that each slice is conformal to (i) a Euclidean space, (ii) a sphere, or (iii) a hyperbolic space, or (iv) Riemannian product of an open interval and and a Riemannian space. Finally, under the preceding hypotheses, if a slice has a K -contact metric, then it is isometric to a unit sphere.