Conformal Transformations between Einstein Spaces

Complete locally conformally flat manifolds of negative curvature

“…Condition (3) is equivalent to the constancy of the sectional curvature of the base of a locally conformally flat multiply warped space. Since (B, g B ) is locally conformally flat and (B, f −2 i g B ) is a space of constant sectional curvature by Lemma 3.1, we see that (B, g B ) is of constant sectional curvature if and only if the conformal deformation g B → f −2 i g B preserves the (unique) eigenspaces of the Ricci tensor, and this occurs if and only if f is a solution of the Möbius equation; this proves (3) (see [Kühnel 1988;Osgood and Stowe 1992]). …”

“…The existence of nontrivial globally defined solutions of (3) on complete manifolds has significant geometrical consequences [Kühnel 1988]. They leads to:…”

“…Since any warping function f defines a global conformal transformation that makes (B, f −2 g B ) have constant curvature, it follows from [Kühnel 1988] that B is either a complete and simply connected space form or a warped product ‫ޒ‬ × αe βt+γ N , where N is complete Ricci flat, and thus flat since B is necessarily locally conformally flat. Now the result will follow after a case by case consideration of the possible warping functions and the curvature of the induced metric f −2 g B .…”

“…A local description of locally conformally flat spaces with the underlying structure of a multiply warped product is then obtained from the fact that any warping function must define a global conformal transformation on the base which makes it of constant sectional curvature. Then the situation when the base has dimension 2 or higher reduces to the existence of nontrivial solutions of some Obata type equations on the base (sometimes called concircular transformations; see [Kühnel 1988;Tashiro 1965]) together with some compatibility conditions among the different warping functions. This analysis is carried out in Section 3A.…”

Einstein Spaces with a Conformal Group

“…[44], [28,Lemma 18] in the Riemannian case) Then there are functions f ± of one variable r such that the metric in geodesic polar coordinates (r, x) valued in R×Σ in a neighborhood U of p has the form…”