  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.
