Equivalence of sub-Laplacian on Polarized groups

TL;DR

This paper characterizes smooth maps between sub-Riemannian Lie groups that commute with sub-Laplacians, revealing they are sub-Riemannian conformal submersions, and shows the sub-Laplacian determines the structure in Carnot groups.

Contribution

It provides a characterization of maps commuting with sub-Laplacians as conformal submersions and clarifies the relationship between sub-Laplacians and sub-Riemannian structures in Carnot groups.

Findings

Maps commuting with sub-Laplacians are conformal submersions

Sub-Laplacian determines the sub-Riemannian structure in Carnot groups

Clarifies analysis on Carnot groups initiated in prior work

Abstract

We characterize smooth maps between sub-Riemannian Lie groups that commute with sub-Laplacians. We show they are sub-Riemannian conformal submersions. Our work clarifies the analysis initiated on Carnot groups in \cite{MR2363343}. In particular, we show that the sub-Laplacian in a Carnot group determines the sub-Riemannian structure.