A Primer on Carnot Groups: Homogenous Groups, Carnot-Carathéodory Spaces, and Regularity of Their Isometries

Enrico Le Donne

doi:10.1515/agms-2017-0007

Abstract

AbstractCarnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance. We present the basic theory of Carnot groups together with several remarks.We consider them as special cases of graded groups and as homogeneous metric spaces.We discuss the regularity of isometries in the general case of Carnot-Carathéodory spaces and of nilpotent metric Lie groups.

Highlights

Introduction to Riemannian and SubRiemannian geometry, Manuscript (2015).[4] Luigi Ambrosio and Bernd Kirchheim, Currents in metric spaces, Acta Math. 185 (2000), no. 1, 1–80. [5], Recti able sets in metric and Banach spaces, Math
Carnot groups are distinguished spaces that are rich of structure: they are those Lie groups equipped with a path distance that is invariant by left-translations of the group and admit automorphisms that are dilations with respect to the distance
We present the basic theory of Carnot groups together with several remarks

Summary

Prototypical examples

First we shall point out that Carnot groups of step one are nite-dimensional normed vector spaces. We are interested in the distances d on V that are (i) translation invariant: d(p + q, p + q ) = d(q, q ), ∀p, q, q ∈ V (ii) one-homogeneous with respect to the dilations: d(λp, λq) = λd(p, q), ∀p, q ∈ V, ∀λ >. Hy) − f (x) h of a function f between vector spaces makes use of the group operation, the dilations, and the topology. We shall consider distances that are (i) left (translation) invariant: d(p · q, p · q ) = d(q, q ), ∀p, q, q ∈ G d(δλ(p), δλ(q)) = λd(p, q), ∀p, q ∈ G, ∀λ >. The space g of left-invariant vector elds ( known as the Lie algebra) has a peculiar structure: it admits a strati cation.

Strati cations

Metric groups

Limits of Riemannian manifolds

Isometrically homogeneous geodesic manifolds

Isometries of metric groups