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–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.


The real Heisenberg group is a Carnot group.


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.

See also[edit]


  • Folland, Gerald (1975), "Subelliptic estimates and function spaces on nilpotent Lie groups", Arkiv for Mat. 13 (2): 161-207.
  • Mitchell, John (1985), "On Carnot-Carathéodory metrics", Journal of Differential Geometry, 21 (1): 35–45, ISSN 0022-040X, MR 0806700
  • Pansu, Pierre (1982), Géometrie du groupe d'Heisenberg, Thesis, Université Paris VII
  • Pansu, Pierre (1989), "Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un", Annals of Mathematics, Second Series, 129 (1): 1–60, doi:10.2307/1971484, ISSN 0003-486X, MR 0979599
  • Bellaïche, André; Risler, Jean-Jacques, eds. (1996). Sub-Riemannian geometry. Progress in Mathematics. 144. Basel: Birkhäuser Verlag. doi:10.1007/978-3-0348-9210-0. MR 1421821.