Spherically symmetric Riemannian manifolds of constant scalar curvature and their conformally flat representations

Abstract

All spherically symmetric Riemannian metrics of constant scalar curvature in any dimension can be written down in a simple form using areal coordinates. All spherical metrics are conformally flat, so we search for the conformally flat representations of these geometries. We find all solutions for the conformal factor in three, four and six dimensions. We write them in closed form, either in terms of elliptic or elementary functions. We are particularly interested in three-dimensional spaces because of the link to General Relativity. In particular, all three-dimensional constant negative scalar curvature spherical manifolds can be embedded as constant mean curvature surfaces in appropriate Schwarzschild solutions. Our approach, although not the simplest one, is linked to the Lichnerowicz–York method of finding initial data for Einstein equations.