On the Geometry of some Para-hypercomplex Lie groups

TL;DR

This paper explores the geometry of para-hypercomplex 4D Lie groups, analyzing their Riemannian and Finsler structures, explicitly calculating connections, curvatures, and constructing Randers metrics with non-positive flag curvature.

Contribution

It provides explicit formulas for Levi-Civita connections, sectional and scalar curvatures, and constructs new Randers metrics of Berwald type on these Lie groups.

Findings

Spaces have constant negative scalar curvature

Explicit formulas for Levi-Civita connection and sectional curvature

Some Finsler metrics exhibit non-positive flag curvature

Abstract

In this paper, firstly we study some left invariant Riemannian metrics on para-hypercomplex 4-dimensional Lie groups. In each Lie group, the Levi-Civita connection and sectional curvature have been given explicitly. We also show these spaces have constant negative scalar curvatures. Then by using left invariant Riemannian metrics introduced in the first part, we construct some left invariant Randers metrics of Berwald type. The explicit formulas for computing flag curvature have been obtained in all cases. Some of these Finsler Lie groups are of non-positive flag curvature.