Carnot-Caratheodory and Koranyi-Geodesics in the Heisenberg Group

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