We consider an overdetermined Steklov eigenvalue problem on a domain in a Riemannian manifold with nonnegative Ricci curvature. We prove that, given a compact connected domain \(\Omega \) with nonnegative Gaussian curvature with \(C^2\) boundary, if the first Steklov eigenfunction is a solution to the overdetermined problem, then the domain \(\Omega \) is flat and the boundary \(\partial \Omega \) consists of geodesics or geodesic circles. This can be regarded as a generalization of the result by Payne-Philippin [21], where they assumed that \(\Omega \) is a simply-connected domain in \(\mathbb {R}^2\) . We also obtain a similar result for the same overdetermined problem on a higher-dimensional compact connected manifold with \(C^2\) boundary with nonnegative Ricci curvature.
