A remark on an overdetermined problem in Riemannian Geometry

Let $(M,g)$ be a Riemannian manifold with a distinguished point $O$ and assume that the geodesic distance $d$ from $O$ is an isoparametric function. Let $\mathrm{\Omega }\subset M$ be a bounded domain, with $O\in \mathrm{\Omega }$, and consider the problem ${\mathrm{\Delta }}_{p}u=-1$ in $\mathrm{\Omega }$ with $u=0$ on $\mathrm{\partial }\mathrm{\Omega }$, where ${\mathrm{\Delta }}_p$ is the $p$-Laplacian of $g$. We prove that if the normal derivative ${\mathrm{\partial }}_{\nu }u$ of $u$ along the boundary of $\mathrm{\Omega }$ is a function of $d$ satisfying suitable conditions, then $\mathrm{\Omega }$ must be a geodesic ball. In particular, our result applies to open balls of ${\mathbb{R}}^n$ equipped with a rotationally symmetric metric of the form $g=d{t}^2+{\rho }^2(t)g_S$, where $g_S$ is the standard metric of the sphere.