ar X iv : g r - qc / 0 31 10 05 v 1 3 N ov 2 00 3 Novel results on trapped surfaces

A unifying definition of trapped submanifold for arbitrary codimension by means of its mean curvature vector is presented. Then, the interplay between (generalized) symmetries and trapped submanifolds is studied, proving in particular that (i) stationary spacetimes cannot contain closed trapped nor marginally trapped sub-manifolds S of any codimension; (ii) S can be within the subset where there is a null Killing vector only if S is marginally trapped with mean curvature vector parallel to the null Killing; (iii) any submanifold orthogonal to a timelike or null Killing vector ξ has a mean curvature vector orthogonal to ξ. All results are purely geometric , hold in arbitrary dimension, and can be appropriately generalized to many non-Killing vector fields, such as conformal Killing vectors and the like. A simple criterion to ascertain the trapping or not of a family of codimension-2 submanifolds is given. A path allowing to generalize the singularity theorems is conjectured as feasible and discussed. As is well-known, the concept of (closed) trapped surface, first introduced by Penrose [1], is of paramount importance in many physical problems and mathematical developments. In particular, it has been an invaluable tool for the development of the singularity theorems (see e.g. [1, 2, 3, 4]), essential for the analysis of gravitational collapse [1, 3, 5], and very useful in the study of the cosmic censorship hypothesis [6] or in the numerical evolution of apparently innocuous initial data (e.g. [7] and references therein.) This contribution is based partly on [8] and partly on [9], where more complete lists of references can be found. Let (V, g) be a D-dimensional causally orientable spacetime with metric g (signature -,+,.. . ,+), Σ any smooth orientable d-dimensional manifold, and Φ : Σ −→ V a C 2 imbedding. Σ and its image Φ(Σ) ≡ S in V will be identified if no confusion 1