Singularity theorems based on trapped submanifolds of arbitrary co-dimension

TL;DR

This paper generalizes singularity theorems in Lorentzian geometry by using trapped submanifolds of any co-dimension and introduces curvature conditions that unify various boundary conditions in classical theorems.

Contribution

It provides a unified framework for singularity theorems applicable to trapped submanifolds of arbitrary co-dimension using mean curvature vectors and sectional curvature conditions.

Findings

Generalized singularity theorems for arbitrary co-dimension

Unified boundary conditions via mean curvature vectors

Replaced classical convergence conditions with sectional curvature criteria

Abstract

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.