Projective relatedness and conformal flatness

Graham Hall

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