The Sasakian Geometry of the Heisenberg Group

TL;DR

This paper explores the Sasakian geometry of the Heisenberg group, showing the Sasaki cone aligns with extremal structures and characterizing when scalar curvature is constant, linking to sub-Riemannian geometry.

Contribution

It establishes the equivalence of the Sasaki cone with extremal structures and characterizes constant scalar curvature metrics in this setting.

Findings

Sasaki cone coincides with extremal Sasakian structures

Constant scalar curvature occurs iff $ ext{Φ}$-sectional curvature is -3

Connections with sub-Riemannian geometry of the Heisenberg group

Abstract

In this note I study the Sasakian geometry associated to the standard CR structure on the Heisenberg group, and prove that the Sasaki cone coincides with the set of extremal Sasakian structures. Moreover, the scalar curvature of these extremal metrics is constant if and only if the metric has $\Phi$-sectional curvature $-3.$ I also briefly discuss some relations with the well-know sub-Riemannian geometry of the Heisenberg group as well as the standard Sasakian structure induced on compact quotients.