Carnot group explained

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 ak{g} admits a step-k stratification. Namely, there exist nontrivial linear subspaces V _{1,} … , V _{k} such that ak{g} = V _{1 ⊕} … ⊕ V _{k} , [V _{1,} V _{i]} = V _{i+1} for i = 1 , … , k -1 , and [V _{1,V} _{k]} = \{ 0 \} .

Note that this definition implies the first stratum

V _{1} generates the whole Lie algebra ak{g} .

The exponential map is a diffeomorphism from

ak{g} onto G . Using these exponential coordinates, we can identify G with ( R ^{n,} \star) , where n = \dim V _{1} + … + \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 asz = (z _{1,} … , z _{k)} with z _{i} \in \R ^{\dim V _{i}} for i = 1 , … , k .

The reason is that

G has an intrinsic dilation operation \delta _{λ} : G \to G given by\delta _{λ(z} _{1,} … , z _{k)} : = ( λ z _{1,} … , λ ^{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.

See also

• Pansu derivative, a derivative on a Carnot group introduced by

References

• • • • • Book: André. Bellaïche . Jean-Jacques . Risler . Sub-Riemannian geometry . Birkhäuser Verlag. Basel . Progress in Mathematics . 144. 1996. 1421821. 10.1007/978-3-0348-9210-0. 978-3-0348-9946-8 .