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

TL;DR

The paper proves that any sectionally positive curvature tensor on an n-dimensional manifold admits a unique (up to scale) Einstein metric tensor, linking curvature conditions to Einstein geometry.

Contribution

It establishes the existence and uniqueness of an Einstein metric tensor for sectionally positive curvature tensors, a novel connection in differential geometry.

Findings

Existence of an Einstein metric tensor for sectionally positive curvature tensors.

Uniqueness of the Einstein metric up to a constant factor.

Extension of Einstein metric existence to a broader class of curvature tensors.

Abstract

Let $n \geq 3$ 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.