We study the Obata equation with Robin boundary condition $\frac{ \partial f}{ \partial \nu}+af=0$ on manifolds with boundary, where $a \in \mathbb{ R}\setminus\{ 0\}$. Dirichlet and Neumann boundary conditions were previously studied by Reilly \cite{ R}, Escobar \cite{ Es} and Xia \cite{ X}. Compared with their results, the sign of $a$ plays an important role here. The new discovery shows besides spherical domains, there are other manifolds for both $a>0$ and $a<0$. We also consider the Obata equation with non-vanishing Neumann condition $\frac{ \partial f}{ \partial \nu}=1$.
Categories: math.DG