Hessian of the Busemann function on Damek-Ricci spaces

TL;DR

This paper computes the Hessian of the Busemann function on Damek-Ricci spaces, analyzes its eigenvalues, and demonstrates its positive definiteness, contributing to the understanding of geometric properties of these spaces.

Contribution

It provides the first explicit calculation of the Hessian of the Busemann function on Damek-Ricci spaces and proves its positive definiteness.

Findings

Hessian of the Busemann function is explicitly calculated.

Eigenvalues of the Hessian are analyzed.

Hessian is proven to be positive definite.

Abstract

In this note, we calculate the Hessian of the Busemann function on a Damek-Ricci space. We investigate the eigenvalues of the Hessian and show its positive definiteness (Theorem 1).