Horizontal curvatures of surfaces in 3D contact sub-Riemannian Lie groups

TL;DR

This paper derives explicit formulas for various horizontal curvatures of surfaces in 3D contact sub-Riemannian Lie groups, focusing on the Heisenberg and affine-additive groups, and classifies revolution surfaces with constant curvatures.

Contribution

It introduces a Riemannian approximation scheme to compute horizontal curvatures and classifies revolution surfaces with constant curvatures in specific sub-Riemannian Lie groups.

Findings

Explicit formulas for horizontal Gauss and mean curvatures

Classification of revolution surfaces with constant curvatures

Profiles expressed through elementary or elliptic integrals

Abstract

In this paper we study horizontal curvatures for surfaces embedded in three-dimensional contact sub-Riemannian Lie groups. Using a Riemannian approximation scheme, we derive explicit formulas for horizontal Gauss curvature, horizontal mean curvature, and symplectic distortion for surfaces embedded in three dimensional Lie groups with a sub-Riemannian structure obtained by a contact form. We focus on two primary examples: the Heisenberg group and the affine-additive group. We classify surfaces of revolution within these groups that exhibit constant horizontal curvatures, often expressing their profiles through elementary or elliptic integrals.