A note on Hang-Wang's hemisphere rigidity theorem

Abstract

Let $(M,g)$ be a compact manifold with boundary and $Ri{c}_g\ge (n-1)g$, Hang and Wang proved that $(M,g)$ is isometric to the standard hemisphere if $\partial M$ is convex and isometric to ${\mathbb{S}}^{n-1}(1)$. We prove some rigidity theorems when $\partial M$ is isometric to a product manifold where one factor is the standard sphere.