# Infinite geodesics and isometric embeddings in Carnot groups of step 2

## TL;DR

This paper proves that in step 2 sub-Finsler Carnot groups with strictly convex norms, all infinite geodesics are lines, and extends this to isometric embeddings being affine, using asymptotic analysis of extremals.

## Contribution

It establishes that all infinite geodesics are lines in certain Carnot groups and characterizes isometric embeddings as affine maps, advancing understanding of geometric structures in these groups.

## Key findings

- All infinite geodesics are lines in step 2 sub-Finsler Carnot groups with strictly convex norms.
- Isometric embeddings between such groups are necessarily affine.
- Geodesic behavior is characterized by asymptotic analysis of extremals via Pontryagin Maximum Principle.

## Abstract

In the setting of step 2 sub-Finsler Carnot groups with strictly convex norms, we prove that all infinite geodesics are lines. It follows that for any other homogeneous distance, all geodesics are lines exactly when the induced norm on the horizontal space is strictly convex. As a further consequence, we show that all isometric embeddings between such homogeneous groups are affine. The core of the proof is an asymptotic study of the extremals given by the Pontryagin Maximum Principle.