Sectionally positive curvature tensors admit a metric tensor under which they are Einstein

Abstract

Let $n\ge 3$ and $R_{abcd}$ be a $(4,0)$ sectionally positive curvature-type tensor (a tensor possessing all the local symmetries of the $(4,0)$ curvature tensor). Then there exists a metric tensor $g_{ab}$ such that $R_{abcd}{g}^{bd}=g_{ac}\lambda$ for some $\lambda$. Furthermore, $g_{ab}$ is unique up to a constant factor.