On the distributional Hessian of the distance function

Carlo Mantegazza, Giovanni Mascellani and Gennady Uraltsev

Vol. 270 (2014), No. 1, 151–166

DOI: 10.2140/pjm.2014.270.151

Abstract

We describe the precise structure of the distributional Hessian of the distance function from a point of a Riemannian manifold. At the same time we discuss some geometrical properties of the cut locus of a point, and compare some different weak notions of the Hessian and Laplacian.

Keywords

distance function, cut locus

Mathematical Subject Classification 2010

Primary: 58E10

Secondary: 35D99

Milestones

Received: 23 April 2013

Revised: 15 January 2014

Accepted: 21 January 2014

Published: 2 August 2014