Pogorelov type interior $C^2$ estimate for sum Hessian quotient equations on Riemannian manifolds

In this paper, we mainly study the interior $C^2$ estimates for a class of sum Hessian quotient equations on Riemannian manifolds. For $0\le l<k<n$, we establish interior $C^2$ estimates at the center of a geodesic ball. Let $\mathrm{\Omega }$ be a bounded domain (with smooth boundary) on Riemannian manifold. For $0\le l<k\le n$, we establish Pogorelov type estimates on $\mathrm{\Omega }$ with the vanishing Dirichlet boundary condition.