Carnot group

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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-Caratheodory 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 subRiemannian manifolds.


The real Heisenberg group is a Carnot group.


Carnot groups were introduced by Pansu (1982, 1989) and Mitchell (1985).

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