Geodesic
On metrics of positive Ricci curvature conformal to $M\times \mathbf{R}^m$
Archivum mathematicum, Tome 45 (2009) no. 2, pp. 105-113

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Let $(M^n,g)$ be a closed Riemannian manifold and $g_E$ the Euclidean metric. We show that for $m>1$, $\left(M^n \times \mathbf{R}^m, (g+g_E)\right)$ 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.
Classification : 53A30, 53C21, 53C25
Keywords: conformally Einstein manifolds; positive Ricci curvature 
@article{ARM_2009__45_2_a3,
     author = {Ruiz, Juan Miguel},
     title = {On metrics of positive {Ricci} curvature conformal to $M\times \mathbf{R}^m$},
     journal = {Archivum mathematicum},
     pages = {105--113},
     publisher = {mathdoc},
     volume = {45},
     number = {2},
     year = {2009},
     mrnumber = {2591667},
     zbl = {1212.53015},
     language = {en},
     url = {http://geodesic.mathdoc.fr/item/ARM_2009__45_2_a3/}
}
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Ruiz, Juan Miguel. On metrics of positive Ricci curvature conformal to $M\times \mathbf{R}^m$. Archivum mathematicum, Tome 45 (2009) no. 2, pp. 105-113. http://geodesic.mathdoc.fr/item/ARM_2009__45_2_a3/