On the Distributional Hessian of the Distance Function

TL;DR

This paper characterizes the distributional Hessian of the distance function on Riemannian manifolds, exploring geometric properties of the cutlocus and comparing various weak notions of Hessian and Laplacian.

Contribution

It provides a detailed description of the distributional Hessian of the distance function and analyzes related geometric and analytical properties.

Findings

Explicit structure of the distributional Hessian derived

Insights into the geometry of the cutlocus

Comparison of weak Hessian and Laplacian notions

Abstract

We describe the precise structure of the distributional Hessian of the distance function from a point of a Riemannian manifold. In doing this we also discuss some geometrical properties of the cutlocus of a point and we compare some different weak notions of Hessian and Laplacian.