  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.
