On totally umbilic submanifolds of semi-Riemannian manifolds

The notion of being totally umbilic is considered for non-degenerate and degenerate submanifolds of semi-Riemannian 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. (C) 2005 Elsevier Ltd. All rights reserved.