Isometries of Carnot groups and subFinsler homogeneous manifolds

TL;DR

This paper proves that isometries in Carnot groups are affine and explores how global isometries of homogeneous manifolds are determined by local behavior, advancing understanding of geometric symmetries in subFinsler spaces.

Contribution

It generalizes Hamenstadt's result by showing isometries are affine in Carnot groups and demonstrates that global isometries of homogeneous manifolds are uniquely determined by their local blow-up.

Findings

Isometries between open sets of Carnot groups are affine.

Global isometries are determined by local blow-up at one point.

Uses Killing vector fields and smoothness results to analyze isometries.

Abstract

We show that isometries between open sets of Carnot groups are affine. This result generalizes a result of Hamenstadt. Our proof does not rely on her proof. In addition, we study global isometries of general homogeneous manifolds equipped with left-invariant subFinsler distances. We show that each isometry is determined by the blow up at one point. For proving the results, we consider the action of isometries on the space of Killing vector fields. We make use of results by Capogna-Cowling and by Gleason-Montgomery-Zippin for obtaining smoothness of the isometric action.