# Geodetic Convexity in the Heisenberg Group

## Authors

Jyotshana V Prajapat
Department of Mathematics, University of Mumbai, Vidyanagari, Mumbai
Anoop Varghese
Department of Mathematics, SIESASCS, Sion (W), Mumbai

### Synopsis

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$$.

Published
February 29, 2024
Series