  In this paper, we establish Pogorelov type $C^2$ estimates for semi-convex
admissible solutions to the Dirichlet problem of $k$-Hessian equation with
general right hand side. Under some sufficient conditions, we apply such
estimates to obtain rigidity theorems for semi-convex admissible solutions of
$k$-Hessian equation, which can be seen as a improvement of Li-Ren-Wang and
Chu-Dinew's rigidity theorem for $k$-Hessian equation. When $2k>n$, we also
obtain Pogorelov type $C^2$ estimates for admissible solutions to the Dirichlet
problem of $k$-Hessian equation based on a concavity inequality, which is
inspired by the Ren-Wang's work on the global curvature estimates for the $n-1$
and $n-2$ curvature equation.
