Geodetic Convexity in the Heisenberg Group

A Heisenberg group \({\mathbb{H}}^n\)(where \(n\ge 1\)) is a Lie group \({\mathbb{C}}^n\mathrm{\times }\mathbb{R}\mathrm{=}\mathrm{\{}\mathrm{(}z,t)|z\in {\mathbb{C}}^n,\ t\in \mathbb{R}\mathrm{\}}\) together with the group operation defined as \(\left(z,t\right)\left(w,s\right)=\left(z+w,t+s+2\mathrm{Im}\left(z.\overline{w}\right)\right)\) where \(z.\overline{w}=\sum^n_{j=1}{z_j\overline{w_j}}\ \) is the Hermitian inner product in the complex space . This group structure imposes constraints on motions in the space \({\mathbb{H}}^n\) giving rise to a geometry which is sub-Riemannian but not Riemannian. Various notions of convex sets have been studied in the Heisenberg Group \({\mathbb{H}}^n\) (for \(n\ge 1\)) which may not necessarily be equivalent. A few of them include horizontal convexity, group convexity, convex in the viscosity sense and geodetic convexity. Here, we discuss the concept of geodetically convex sets in \({\mathbb{H}}^n\) for \(n\ge 1\) and classify them. A geodetically convex set in \({\mathbb{H}}^n\) is defined to be a set which contains every geodesic connecting every pair of points in the set. We prove that every geodetically convex set in \({\mathbb{H}}^n\) is either an empty set, a singleton set, an arc of a geodesic or the whole space \({\mathbb{H}}^n\). These results generalise the known results of \({\mathbb{H}}^{\mathrm{1}}\) to \({\mathbb{H}}^n\) for \(n\ge 1\).