Carnot group
In mathematics, a Carnot group is a simply connected nilpotent Lie group, together with a derivation of its Lie algebra such that the subspace with eigenvalue 1 generates the Lie algebra. The subbundle of the tangent bundle associated to this eigenspace is called horizontal. On a Carnot group, any norm on the horizontal subbundle gives rise to a Carnot–Carathéodory metric. Carnot–Carathéodory metrics have metric dilations; they are asymptotic cones (see Ultralimit) of finitely-generated nilpotent groups, and of nilpotent Lie groups, as well as tangent cones of sub-Riemannian manifolds. Formal definition and basic properties

A Carnot (or stratified) group of step  is a connected, simply connected, finite-dimensional Lie group whose Lie algebra  admits a step- stratification. Namely, there exist nontrivial linear subspaces  such that 

,  for , and . Note that this definition implies the first stratum  generates the whole Lie algebra . The exponential map is a diffeomorphism from  onto . Using these exponential coordinates, we can identify  with , where  and the operation  is given by the Baker–Campbell–Hausdorff formula. Sometimes it is more convenient to write an element  as

 with  for . The reason is that  has an intrinsic dilation operation  given by

. Examples

The real Heisenberg group is a Carnot group which can be viewed as a flat model in Sub-Riemannian geometry as Euclidean space in Riemannian geometry. The Engel group is also a Carnot group. History 
Carnot groups were introduced, under that name, by  and . However, the concept was introduced earlier by Gerald Folland (1975), under the name stratified group. See also
Pansu derivative, a derivative on a Carnot group introduced by

References

Lie groups