  In this paper, we characterize the rigidity of umbilical hypersurfaces by a
Serrin-type partially overdetermined problem in space forms, which generalizes
the similar results in Euclidean half-space and Euclidean half-ball. Guo-Xia
first obtained these rigidity results when the Robin boundary condition on the
support hypersurface is homogeneous, at this time the target umbilical
hypersurface has orthogonal contact angle with the support. However, in this
paper we can obtain any contact angle $\theta\in (0,\pi)$ by changing the Robin
boundary condition to be inhomogeneous.
