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Keywords:
conformally Einstein manifolds; positive Ricci curvature
conformally Einstein manifolds; positive Ricci curvature
Summary:
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.
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