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.
A Carnot (or stratified) group of step
is a connected, simply connected, finite-dimensional Lie group whose Lie algebra
Namely, there exist nontrivial linear subspaces
such that Note that this definition implies the first stratum
generates the whole Lie algebra
given by 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.
Carnot groups were introduced, under that name, by Pierre Pansu (1982, 1989) and John Mitchell (1985).
However, the concept was introduced earlier by Gerald Folland (1975), under the name stratified group.
This abstract algebra-related article is a stub.