Hessian of Busemann functions and rank of Hadamard manifolds

TL;DR

This paper investigates the Hessian of Busemann functions on harmonic Hadamard manifolds, showing positive definiteness under certain conditions and relating it to geodesic rank and visibility properties.

Contribution

It establishes the positive definiteness of Busemann Hessians on harmonic Hadamard manifolds and links this to geodesic rank and visibility criteria.

Findings

Hessian of Busemann functions is positive definite on harmonic Damek-Ricci spaces.

On certain harmonic Hadamard manifolds, all Busemann functions have positive definite Hessians.

A criterion for the visibility axiom based on Hessian positive definiteness is provided.

Abstract

In this article we show that every geodesic is rank one and the Hessian of Busemann functions is positive definite for a harmonic Damek-Ricci space, a two step solvable Lie group with a left invariant metric. Moreover, the eigenspace of the Hessian of Busemann functions on a Hadamard manifold $(M,g)$ corresponding to eigenvalue zero is investigated with respect to rank of geodesics. On a harmonic Hadamard manifold which is of purely exponential volume growth, or of hypergeometric type it is shown that every Busemann function admits positive definite Hessian. A criterion for $(M,g)$ fulfilling visibility axiom is presented in terms of positive definiteness of the Hessian of Busemann functions.