Overdetermined problems for fully nonlinear equations in space forms

We consider overdetermined problems for a class of fully nonlinear equations with constant boundary conditions in a bounded domain in space forms. If the domain is star-shaped, the solution to the Hessian quotient overdetermined problem is radial symmetric. By establishing a Rellich-Poho\v{z}aev type identity for $k$-Hessian equations with constant boundary conditions, we also show the symmetry of solutions to $k$-Hessian equations without assuming the star-shapedness of the domain.