  Let $(M,g)$ be a compact manifold with boundary and $Ric_g\geq (n-1)g$, 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.
