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 ${\displaystyle k}$ is a connected, simply connected, finite-dimensional Lie group whose Lie algebra ${\displaystyle {\mathfrak {g}}}$ admits a step-${\displaystyle k}$ stratification. Namely, there exist nontrivial linear subspaces ${\displaystyle V_{1},\cdots ,V_{k}}$ such that

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

Note that this definition implies the first stratum ${\displaystyle V_{1}}$ generates the whole Lie algebra ${\displaystyle {\mathfrak {g}}}$.

The exponential map is a diffeomorphism from ${\displaystyle {\mathfrak {g}}}$ onto ${\displaystyle G}$. Using these exponential coordinates, we can identify ${\displaystyle G}$ with ${\displaystyle (\mathbb {R} ^{n},\star )}$, where ${\displaystyle n=\dim V_{1}+\cdots +\dim V_{k}}$ and the operation ${\displaystyle \star }$ is given by the Baker–Campbell–Hausdorff formula.

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

${\displaystyle z=(z_{1},\cdots ,z_{k})}$ with ${\displaystyle z_{i}\in \mathbb {R} ^{\dim V_{i}}}$ for ${\displaystyle i=1,\cdots ,k}$.

The reason is that ${\displaystyle G}$ has an intrinsic dilation operation ${\displaystyle \delta _{\lambda }:G\to G}$ given by

${\displaystyle \delta _{\lambda }(z_{1},\cdots ,z_{k}):=(\lambda z_{1},\cdots ,\lambda ^{k}z_{k})}$.

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 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

• Pansu derivative, a derivative on a Carnot group introduced by Pansu (1989)

References

• Folland, Gerald (1975), "Subelliptic estimates and function spaces on nilpotent Lie groups", Arkiv för Matematik, 13 (2): 161–207, Bibcode: 1975ArM....13..161F, doi: 10.1007/BF02386204, S2CID  121144337
• Mitchell, John (1985), "On Carnot-Carathéodory metrics", Journal of Differential Geometry, 21 (1): 35–45, doi: 10.4310/jdg/1214439462, ISSN  0022-040X, MR  0806700
• Pansu, Pierre (1982), Géometrie du groupe d'Heisenberg, Thesis, Université Paris VII{{ citation}}: CS1 maint: location missing publisher ( link)
• Pansu, Pierre (1989), "Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un", Annals of Mathematics, 129 (1): 1–60, doi: 10.2307/1971484, JSTOR  1971484, MR  0979599
• Bellaïche, André; Risler, Jean-Jacques, eds. (1996). Sub-Riemannian geometry. Progress in Mathematics. Vol. 144. Basel: Birkhäuser Verlag. doi: 10.1007/978-3-0348-9210-0. ISBN  978-3-0348-9946-8. MR  1421821.

This abstract algebra-related article is a stub. You can help Wikipedia by expanding it.

• v
• t
• e