| 释义 |
- Examples
- History
- See also
- References
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. ExamplesThe real Heisenberg group is a Carnot group. HistoryCarnot groups were introduced, under that name, by {{harvs|txt|last=Pansu|first=Pierre|authorlink=Pierre Pansu|year1=1982|year2=1989}} and {{harvs|txt|first=John|last=Mitchell|year=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 {{harvtxt|Pansu|1989}}
References- Folland, Gerald (1975), "Subelliptic estimates and function spaces on nilpotent Lie groups", Arkiv for Mat. 13 (2): 161-207.
- {{Citation | last1=Mitchell | first1=John | title=On Carnot-Carathéodory metrics | url=http://projecteuclid.org/getRecord?id=euclid.jdg/1214439462 | mr=806700 | year=1985 | journal=Journal of Differential Geometry | issn=0022-040X | volume=21 | issue=1 | pages=35–45}}
- {{citation|last=Pansu | first=Pierre |authorlink=Pierre Pansu| title=Géometrie du groupe d'Heisenberg|series=Thesis|place=Université Paris VII|year=1982|url=http://www.math.u-psud.fr/~pansu/pansu_These_1982.html}}
- {{Citation | last1=Pansu | first1=Pierre |authorlink=Pierre Pansu | title=Métriques de Carnot-Carathéodory et quasiisométries des espaces symétriques de rang un | doi=10.2307/1971484 | mr=979599 | year=1989 | journal=Annals of Mathematics |series=Second Series | issn=0003-486X | volume=129 | issue=1 | pages=1–60}}
- {{cite book | editor1-first=André|editor1-last=Bellaïche | editor2-first=Jean-Jacques | editor2-last=Risler | title=Sub-Riemannian geometry | url=https://www.springer.com/gb/book/9783764354763 | publisher=Birkhäuser Verlag| location=Basel |series = Progress in Mathematics |volume=144| year = 1996|mr=1421821|doi=10.1007/978-3-0348-9210-0}}
{{abstract-algebra-stub}} 1 : Lie groups |