  Standard singularity theorems are proven in Lorentzian manifolds of arbitrary
dimension n if they contain closed trapped submanifolds of arbitrary
co-dimension. By using the mean curvature vector to characterize trapped
submanifolds, a unification of the several possibilities for the boundary
conditions in the traditional theorems and their generalization to arbitrary
co-dimension is achieved. The classical convergence conditions must be replaced
by a condition on sectional curvatures, or tidal forces, which reduces to the
former in the cases of co-dimension 1, 2 or n.

General Relativity and Quantum Cosmology