## 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.

Original language | English (US) |
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Pages (from-to) | 1-12 |

Number of pages | 12 |

Journal | Journal of Geometry and Physics |

Volume | 41 |

Issue number | 1-2 |

DOIs | |

State | Published - Feb 2002 |

## Keywords

- Hausdorff measure
- Horizons
- Riemannian geometry

## ASJC Scopus subject areas

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