Photon surfaces with equipotential time slices

Photon surfaces are timelike totally umbilic hypersurfaces of Lorentzian spacetimes. In the first part of this paper, we locally characterize all possible photon surfaces in a class of static spherically symmetric spacetimes that includes (exterior) Schwarzschild, Reissner-Nordstrom, and Schwarzschild-anti de Sitter in n + 1 dimensions. In the second part, we prove that any static, vacuum, and "asymptotically isotropic" n + 1-dimensional spacetime that possesses what we call an "equipotential" and "outward directed" photon surface is isometric to the Schwarzschild spacetime of the same (necessarily positive) mass using a uniqueness result obtained by the first author.