Carnot-Caratheodory and Koranyi-Geodesics in the Heisenberg Group

TL;DR

This paper explores the properties of the Heisenberg group with two equivalent metrics, demonstrating that the lengths of curves are consistent across these metrics and establishing the existence of geodesics.

Contribution

It shows the equivalence of curve lengths and proves the existence of geodesics in the Heisenberg group for both Koranyi and Carnot-Caratheodory metrics.

Findings

Curve lengths coincide for both metrics

Shortest curves, geodesics, exist in the Heisenberg group

Metrics are equivalent in defining curve length

Abstract

This paper is part of an undergraduate research project. We discuss the Heisenberg group H1, the three-dimensional space R3 equipped with one of two equivalent metrics, the Koranyi- and Carnot- Caratheodory metric. We show that the notion of length of curves for both metrics coincide, and that shortest curves, so-called geodesics, exist.