Rigidity of positively curved vacuum Static spaces

Abstract: In this paper, we study compact vacuum static spaces of dimension $\ge 4$ with nonnegative sectional curvature. First, we prove that an $n$-dimensional compact vacuum static space is isometric to a standard sphere provided it has positive sectional curvature and the complete divergence of its Weyl curvature tensor is nonnegative. Under the slightly weaker condition of nonnegative sectional curvature, we show that such a vacuum static space must have parallel Ricci curvature tensor. Second, we examine the critical point equation arising from the critical metrics of the total scalar curvature functional on the space of Riemannian metrics restricted to unit volume and constant scalar curvature. We demonstrate that a compact Riemannian manifold with nonnegative sectional curvature and nonnegative complete divergence of the Weyl curvature tensor, which admits a nontrivial solution to the critical point equation, must be Einstein and is isometric to a standard sphere. Our results can be viewed as extensions of the results in [4].