Reissner-Nordstr{\o}m-de Sitter Manifold : Photon Sphere and Maximal Analytic Extension

This paper is devoted to the study of the Reissner-Nordstr{\o}m-de Sitter black holes and their maximal analytic extensions. In particular, we study some of their properties that lays the groundwork for separate papers where we obtain decay results and construct conformal scattering theories for test fields on such spacetimes. Here, we find the necessary and sufficient conditions on the parameters of the Reissner-Nordstr{\o}m-de Sitter metric -namely, the mass, the charge, and the cosmological constant- to have three horizons. Under this conditions, we prove that there is only one photon sphere and we locate it. We then give a detailed construction of the maximal analytic extension of the Reissner-Nordstr{\o}m-de Sitter manifold in the case of three horizons.