Photon surfaces with equipotential time slices

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