When are Carnot groups complete and negatively curved (in the sense of [$CAT(\kappa)$](https://en.wikipedia.org/wiki/CAT(k)_space) spaces)? 



When are Carnot groups negatively curved and homeomorphic to Euclidean space
All Carnot groups are complete metric spaces, since they have all closed balls compact ("proper" metric space). In general, any metric space with a transitive isometry group, and having a compact subset with nonempty interior, is complete (easy exercise).


The result that every Carnot group of dimension $\ge 2$ is not CAT($−\kappa$) for any $\kappa>0$ is straightforward. Indeed, since it has a non-isometric self-homothety, it would imply that it is CAT($−\kappa'$) for every $\kappa'>0$, and hence CAT($-\infty$), which for a geodesic space means an $\mathbf{R}$-tree, which cannot have any subset homeomorphic to the plane.


Actually, a non-abelian Carnot group is not even CAT(0), and does not even have a quasi-isometric embedding into any CAT(0) space (or any uniformly convex Banach space). The latter fact was established in:


Scott D. Pauls. The large scale geometry in nilpotent Lie groups. Commun. Anal. Geom. 9(5), 951-982, 2001. However, 


