On metrics of positive Ricci curvature conformal to MxR^m

TL;DR

This paper proves that certain product manifolds with a conformal Euclidean factor cannot be conformally transformed into manifolds with positive Einstein or Ricci curvature, addressing questions related to Yamabe constants.

Contribution

It establishes non-existence results for conformal transformations to positive Ricci or Einstein metrics on product manifolds with Euclidean factors for dimensions greater than one.

Findings

Product manifolds with Euclidean factors are not conformally equivalent to positive Einstein manifolds.

Such manifolds cannot be conformally transformed into positive Ricci curvature manifolds via smooth radial functions.

Results contribute to understanding Yamabe constant-related questions.

Abstract

Let (M, g) be a closed Riemannian manifold and gE the Euclidean metric. We show that for m > 1, (M x R^m, (g + gE)) is not conformal to a positive Einstein manifold. Moreover, (M x R^m, (g + gE)) is not conformal to a Riemannian manifold of positive Ricci curvature, through a smooth, radial, positive, integrable function of R^m, for m > 1. These results are motivated by some recent questions on Yamabe constants.