  In this work, we are interested in studying Serrin's overdetermined problems
in Riemannian manifolds. For manifolds endowed with a conformal vector field,
we prove a Pohozoaev-type identity to show a Serrin's type rigidity result
using the P-function approach introduced by Weinberger. We proceed with a
conformal change to achieve this goal, starting from a geometric Pohozaev
identity due to Schoen. Moreover, we obtain a symmetry result for the
associated Dirichlet problem by using a generalized normalized wall shear
stress bound.
