On fine differentiability properties of horizons and applications to Riemannian geometry

Piotr T. Chruściel, Joseph H.G. Fu, Gregory J. Galloway, Ralph Howard

Research output: Contribution to journalArticlepeer-review

12 Scopus citations

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 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.

Original languageEnglish (US)
Pages (from-to)1-12
Number of pages12
JournalJournal of Geometry and Physics
Volume41
Issue number1-2
DOIs
StatePublished - Feb 2002

Keywords

  • Hausdorff measure
  • Horizons
  • Riemannian geometry

ASJC Scopus subject areas

  • Mathematical Physics
  • Physics and Astronomy(all)
  • Geometry and Topology

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