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 $$k$$ is a connected, simply connected, finite-dimensional Lie group whose Lie algebra $$\mathfrak{g}$$ admits a step-$$k$$ stratification. Namely, there exist nontrivial linear subspaces $$V_1, \cdots, V_k$$ such that


 * $$\mathfrak{g} = V_1\oplus \cdots \oplus V_k$$, $$[V_1, V_i] = V_{i+1}$$ for $$i = 1, \cdots, k-1$$, and $$[V_1,V_k] = \{0\}$$.

Note that this definition implies the first stratum $$V_1$$ generates the whole Lie algebra $$\mathfrak{g}$$.

The exponential map is a diffeomorphism from $$\mathfrak{g}$$ onto $$G$$. Using these exponential coordinates, we can identify $$G$$ with $$(\mathbb{R}^n, \star)$$, where $$n = \dim V_1 + \cdots + \dim V_k $$ and the operation $$\star$$ is given by the Baker–Campbell–Hausdorff formula.

Sometimes it is more convenient to write an element $$z \in G$$ as


 * $$z = (z_1, \cdots, z_k)$$ with $$z_i \in \R^{\dim V_i}$$ for $$i = 1, \cdots, k$$.

The reason is that $$G$$ has an intrinsic dilation operation $$\delta_\lambda : G \to G$$ given by


 * $$\delta_\lambda(z_1, \cdots, z_k) := (\lambda z_1, \cdots, \lambda^k z_k)$$.

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.