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

TL;DR

This paper classifies spherically symmetric Riemannian manifolds with constant scalar curvature and finds their conformally flat representations in dimensions 3, 4, and 6, with implications for General Relativity.

Contribution

It provides explicit solutions for conformal factors in multiple dimensions, linking geometric structures to initial data in Einstein's equations.

Findings

Explicit conformal factors in 3, 4, and 6 dimensions

All spherical metrics of constant scalar curvature are conformally flat

3D negative scalar curvature manifolds embed in Schwarzschild solutions

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 3, 4 and 6 dimensions. We write them in closed form, either in terms of elliptic or elementary functions. We are particularly interested in 3-dimensional spaces because of the link to General Relativity. In particular, all 3-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.