Trapped submanifolds in Lorentzian geometry

Abstract

The concept of closed trapped surface is of paramount importance in General Relativity and other gravitational theories. However, it is a purely geometrical object. With the aim of bringing this concept to closer attention by the mathematical community, I introduce the generalized idea of trapped submanifold by using the causal character of the mean curvature vector. Trapped surfaces can thus be easily seen to be generalizations, possible in semi-Riemannian geometry, of the standard concepts of minimal surfaces, maximal hypersurfaces, geodesics, and the like. Selected applications are mentioned.

Contribution to the proceedings of the XIII Fall Workshop on Geometry and Physics, (Murcia, Spain, 2004); Introduction added and minor corrections made