  We mainly study Pogorelov type $C^2$ estimates for solutions to the Dirichlet
problem of Sum Hessian equations. We establish respectively Pogorelov type
$C^2$ estimates for $k$-convex solutions and admissible solutions under some
conditions. Furthermore, we apply such estimates to obtain a rigidity theorem
for $k$-convex solutions of Sum Hessian equations in Euclidean space.
