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 a n-dimensional horizon H (which is included in a (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 C2 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 C2 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.
Document Details
- Document Type
- Technical Report
- Publication Date
- Nov 21, 2000
- Accession Number
- ADA640707
Entities
People
- Gregory J. Galloway
- Joseph H. Fu
- Piotr T. Chrusciel
- Ralph Howard
Organizations
- University of South Carolina