On fine differentiability properties of horizons and applications to Riemannian geometry

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 less than or equal to k less than or equal to 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 + I - 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. (C) 2002 Elsevier Science B.V. All rights reserved.