On some hypercomplex 4-dimensional Lie groups of constant scalar curvature

TL;DR

This paper investigates the sectional curvature of invariant hyper-Hermitian metrics on 4D Lie groups with hypercomplex structures, providing explicit formulas and showing these spaces have constant scalar curvature, being either flat or with non-negative/non-positive sectional curvature.

Contribution

It offers explicit formulas for Levi-Civita connections and sectional curvatures on these Lie groups, and classifies their curvature properties.

Findings

All studied spaces have constant scalar curvature.

They are either flat or have sectional curvature of only non-negative or non-positive values.

Explicit formulas for Levi-Civita connections are provided.

Abstract

In this paper we study sectional curvature of invariant hyper-Hermitian metrics on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. We give the Levi-Civita connections and explicit formulas for computing sectional curvatures of these metrics and show that all these spaces have constant scalar curvature. We also show that they are flat or they have only non-negative or non-positive sectional curvature.