  The notion of being totally umbilic is considered for non-degenerate and
degenerate submanifolds of semi-Riemanian manifolds. After some remarks on the
general case, timelike and lightlike totally umbilic submanifolds of Lorentzian
manifolds are discussed, along with their physical interpretation in view of
general relativity. In particular, the mathematical notion of totally umbilic
submanifolds is linked to the notions of photon surfaces and of null strings
which have been used in the physics literature.
