Infinite geodesics of sub-Finsler distances in the Heisenberg groups

TL;DR

This paper proves that in Heisenberg groups with sub-Finsler metrics defined by strictly convex norms, all infinite geodesics are horizontal lines, advancing understanding of sub-Finsler geometry and isometric embeddings.

Contribution

It establishes that infinite geodesics are horizontal lines in this setting, answering a previously posed question and aiding in characterizing isometric embeddings.

Findings

Infinite geodesics are horizontal lines under the given conditions.

The result applies to sub-Finsler metrics defined by strictly convex norms.

The work has implications for isometric embedding characterizations.

Abstract

We consider Heisenberg groups equipped with a sub-Finsler metric. Using methods of optimal control theory we prove that in this geometric setting the infinite geodesics are horizontal lines under the assumption that the sub-Finsler metric is defined by a strictly convex norm. This answers a question posed in [5] and has applications in the characterisation of isometric embeddings into Heisenberg groups.