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