  We consider overdetermined problems for two classes of fully nonlinear
equations with constant Dirichlet boundary conditions in a bounded domain in
space forms. We prove that if the domain is star-shaped, then the solution to
the Hessian quotient overdetermined problem is radially symmetric. By
establishing a Rellich-Poho\v{z}aev type identity for the $k$-Hessian equation
with constant Dirichlet boundary condition, we also show the radial symmetry of
the solution to the $k$-Hessian overdetermined problem for some boundary value
without star-shapedness assumption of the domain.
