Horizon classification via Riemannian flows
R. A. Hounnonkpe, E. Minguzzi
TL;DR
This paper reframes compact horizons in Lorentzian spacetimes as Riemannian flows on horizons, enabling the import of Molino–Carrière-type results to horizon dynamics without assuming non-degeneracy. It defines a transverse Levi-Civita connection and shows horizons inherit a rich Riemannian-flow structure, including isometric and transverse Killing-field symmetries, with implications for horizon classification. In 4D, the authors recover the Bustamante–Reiris classification for non-degenerate horizons under the dominant energy condition and identify an additional $\mathbb{T}^3_A$ hyperbolic structure in the degenerate case; they also establish conditions under which surface gravity can be set to zero, via circle-bundle constructions and Diophantine-flow arguments, and prove vanishing of $d\omega$ in the dense-generator non-degenerate regime. Overall, the work bridges horizon geometry with the theory of Riemannian foliations to derive structural results, symmetry properties, and zero-surface-gravity scenarios with potential extensions to higher dimensions.
Abstract
We point out that the geometry of connected totally geodesic compact null hypersurfaces in Lorentzian manifolds is only slightly more specialized than that of Riemannian flows over compact manifolds, the latter mathematical theory having been much studied in the context of foliation theory since the work by Reinhart (Ann Math 69:119, 1959). We are then able to import results on Riemannian flows to the horizon case, so obtaining theorems on the dynamical structure of compact horizons that do not rely on (non-)degeneracy assumptions. Furthermore, we clarify the relation between isometric/geodesible Riemannian flows and non-degeneracy conditions. This work also contains some positive results on the possibility of finding, in the degenerate case, lightlike fields tangent to the horizon that have zero surface gravity.