The Obata equation with Robin boundary condition

We study the Obata equation with Robin boundary condition \partial f/\partial \nu + af = 0 on manifolds with boundary, where a is a non-zero constant. Dirichlet and Neumann boundary conditions were previously studied by Reilly, Escobar and Xia. 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 \partial f/\partial \nu=1 .