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 $(V,g,\mathcal K)$ (with metric $g$ and extrinsic curvature ${\mathcal K}$) which is 2-convex (meaning that the sum of the lowest two eigenvalues of ${\mathcal K}$ is non-negative), then either $M$ is past null geodesically incomplete, or $V$ is a spherical space, or $V$ or some finite cover $\tilde V$ is a surface bundle over the circle, with totally geodesic fibers. Moreover, (cf. Theorem 2) if $(V,g,{\mathcal K})$ admits a $U(1)$ isometry group with corresponding Killing vector $\xi$, we can relax the convexity requirement in terms of a decomposition of ${\mathcal K}$ with respect to the directions parallel and orthogonal to $\xi$. Finally, (cf. Propositions 1-3) in the special cases that $V$ is either nonorientable, or non-prime, or an orientable Haken manifold with vanishing second homology, we obtain stronger statements in both Theorems without passing to covers. While our results do not use Einstein's equations, we provide several classes of $\Lambda$-vacuum solutions of these equations as examples.