On fine differentiability properties of horizons and applications to Riemannian geometry

Abstract

We study fine differentiability properties of horizons. We show that the set of end points of generators of an n-dimensional horizon H (which is included in an ( n+1)-dimensional space–time M) has vanishing n-dimensional Hausdorff measure. This is proved by showing that the set of end points of generators at which the horizon is differentiable has the same property. For 1≤ k≤ n+1, we show (using deep results of Alberti) that the set of points where the convex hull of the set of generators leaving the horizon has dimension k is “almost a C 2 manifold of dimension n+1− k”: it can be covered, up to a set of vanishing ( n+1− k)-dimensional Hausdorff measure, by a countable number of C 2 manifolds. We use our Lorentzian geometry results to derive information about the fine differentiability properties of the distance function and the structure of cut loci in Riemannian geometry.