On metrics of positive Ricci curvature conformal to $M\times \mathbf{R}^m$

Juan Miguel Ruiz

Address: CIMAT, Jalisco S/N, Col. Valenciana, CP 36240 Guanajuato Guanajuato Mexico

E-mail: miguel@cimat.mx

Abstract: Let $(M^n,g)$ be a closed Riemannian manifold and $g_E$ the Euclidean metric. We show that for $m>1$, $\left(M^n \times \mathbf{R}^m, (g+g_E)\right)$ is not conformal to a positive Einstein manifold. Moreover, $\left(M^n \times \mathbf{R}^m, (g+g_E)\right)$ is not conformal to a Riemannian manifold of positive Ricci curvature, through a radial, integrable, smooth function, $\varphi \colon \mathbf{R^m} \rightarrow \mathbf{R^+}$, for $m>1$. These results are motivated by some recent questions on Yamabe constants.

AMSclassification: primary 53A30; secondary 53C21, 53C25.

Keywords: conformally Einstein manifolds, positive Ricci curvature.