[Paper Review] Isometries of Carnot groups and subFinsler homogeneous manifolds

Enrico Le Donne, Alessandro Ottazzi|arXiv (Cornell University)|Oct 18, 2012

Advanced Differential Geometry Research16 references5 citations

TL;DR

This paper establishes that isometries between open sets in Carnot groups are affine, generalizing Hamenstadt’s result via a novel proof independent of her approach. It further shows that global isometries on left-invariant subFinsler homogeneous manifolds are determined by their blow-up behavior at a single point, leveraging the action of isometries on Killing vector fields and smoothness results from Capogna-Cowling and Gleason-Montgomery-Zippin.

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.

Motivation & Objective

• To establish that isometries between open subsets of Carnot groups are affine, extending Hamenstadt’s result with a new proof.
• To analyze global isometries on left-invariant subFinsler homogeneous manifolds using geometric and analytic tools.
• To show that each isometry is uniquely determined by its infinitesimal behavior at a single point via blow-up analysis.
• To leverage the action of isometries on the space of Killing vector fields to deduce regularity and structure of isometric actions.
• To apply deep results from differential geometry and Lie group theory—specifically Capogna-Cowling and Gleason-Montgomery-Zippin—to establish smoothness of the isometric action.

Proposed method

• Analyzing the action of isometries on the space of Killing vector fields to extract structural constraints.
• Using blow-up techniques at a single point to reconstruct global isometries on subFinsler homogeneous manifolds.
• Applying theorems by Capogna and Cowling on regularity of isometric actions on subRiemannian manifolds.
• Utilizing results by Gleason, Montgomery, and Zippin on the structure of Lie groups and smooth actions to ensure smoothness of isometric maps.
• Establishing that isometries on Carnot groups are affine by combining symmetry analysis with the blow-up method.
• Proving that the isometric action is smooth by reducing the problem to the behavior of vector fields under isometric transformations.

Experimental results

Research questions

• RQ1 Are isometries between open sets in Carnot groups necessarily affine, and can this be proven without relying on Hamenstadt’s original approach?
• RQ2 To what extent are global isometries on left-invariant subFinsler homogeneous manifolds determined by their local behavior at a single point?
• RQ3 How does the action of isometries on the space of Killing vector fields constrain the global structure of isometric maps?
• RQ4 What role do smoothness theorems from Lie group theory play in establishing regularity of isometric actions on subFinsler manifolds?
• RQ5 Can the blow-up of an isometry at a point fully reconstruct the global isometry in subFinsler homogeneous geometries?

Key findings

• Isometries between open sets in Carnot groups are affine, providing a strong rigidity result independent of Hamenstadt’s original proof.
• Global isometries on left-invariant subFinsler homogeneous manifolds are completely determined by their blow-up at a single point.
• The action of isometries on Killing vector fields is instrumental in deducing the affine structure of isometries on Carnot groups.
• Smoothness of the isometric action is established through the application of Capogna-Cowling and Gleason-Montgomery-Zippin theorems on Lie group actions.
• The blow-up method provides a local-to-global reconstruction mechanism for isometries in subFinsler settings.
• The results extend rigidity and regularity properties from Riemannian to subFinsler geometries in the context of homogeneous spaces.