The Heisenberg Group and Sub-Riemannian Geometry

Abstract

In this chapter we provide a detailed description of the sub-Riemannian geometry of the first Heisenberg group. We describe its algebraic structure, introduce the horizontal subbundle (which we think of as constraints) and present the Carnot-Carathéodory metric as the least time required to travel between two given points at unit speed along horizontal paths. Subsequently we introduce the notion of sub-Riemannian metric and show how it arises from degenerating families of Riemannian metrics. For use in later chapters we compute some of the standard differential geometric apparatus in these Riemannian approximants.