Rigidity aspects of a cosmological singularity theorem

Improving a singularity theorem in General Relativity by Galloway and Ling we show the following (cf.\ Theorem 1): If a globally hyperbolic spacetime M satisfying the null energy condition contains a closed, spacelike Cauchy surface $(V,g,K)$ (with metric g and extrinsic curvature K) which is 2-convex (meaning that the sum of the lowest two eigenvalues of K is non-negative), then either M is past null geodesically incomplete, or V is a spherical space, or V or some finite cover is a surface bundle over the circle, with totally geodesic fibers. Moreover, (cf.\ Theorem 2) if $(V,g,K)$ admits a $U(1)$ isometry group with corresponding Killing vector $ξ$, we can relax the convexity requirement in terms of a decomposition of K with respect to the directions parallel and orthogonal to $ξ$. Finally, (cf. Propositions 1-3) in the special cases that V is either non-orientable, or non-prime, or an orientable Haken manifold with vanishing second homology, we obtain stronger statements in both Theorems without passing to covers.