Conformal maps of Carnot groups

TL;DR

This paper characterizes conformal maps on Carnot groups, showing they are either trivial transformations or arise from specific Lie group actions, revealing the structure of conformal symmetries in these geometric settings.

Contribution

It provides a complete classification of conformal maps on Carnot groups, linking them to Lie group actions and geometric structures, extending previous understanding of conformal geometry in sub-Riemannian contexts.

Findings

Conformal maps are either compositions of translations, dilations, and isometries.

In certain cases, conformal maps originate from actions of rank 1 simple Lie groups.

The classification depends on the structure of the Carnot group and its relation to Lie groups.

Abstract

If f is a conformal mapping defined on a connected open subset of a Carnot group G, then either f is the composition of a translation, a dilation and an isometry, or G is the nilpotent Iwasawa component of a real rank 1 simple Lie group S, and f arises from the action of S on G, viewed as an open subset of S/P, where P is a parabolic subgroup of G and NP is open and dense in S.