  Rank-one symmetric spaces carry a solvable group model which have a
generalization to a larger class of Lie groups that are one-dimensional
extensions of nilpotent groups. By examining some metric properties of these
symmetric spaces, we motivate and prove the existence of analogous
left-invariant, Riemannian metrics on Heintze groups of Carnot-type. These
metrics adhere to certain natural curvature pinching properties, and we show in
a special case that this pinching is optimal, appealing to a result of
Belegradek and Kapovitch.
