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 formula_1 is a connected, simply connected, finite-dimensional Lie group whose Lie algebra formula_2 admits a step-formula_1 stratification. Namely, there exist nontrivial linear subspaces formula_4 such that 
Note that this definition implies the first stratum formula_9 generates the whole Lie algebra formula_2.
The exponential map is a diffeomorphism from formula_2 onto formula_12. Using these exponential coordinates, we can identify formula_12 with formula_14, where formula_15 and the operation formula_16 is given by the Baker–Campbell–Hausdorff formula.
Sometimes it is more convenient to write an element formula_17 as
The reason is that formula_12 has an intrinsic dilation operation formula_22 given by
Examples.
The real Heisenberg group is a Carnot group which can be viewed as a flat model in Sub-Riemannian geometry as Eculidean 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.