# On the geometry of some para-hypercomplex Lie groups

## H. R. Salimi Moghaddam

Address: Department of Mathematics, Shahrood University of Technology, Shahrood, Iran

E-mail:

hrsalimi@shahroodut.ac.ir

salimi.moghaddam@gmail.com

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.

AMSclassification: primary 53C15; secondary 58B20, 53B35, 53C60.

Keywords: para-hypercomplex structure, left invariant Riemannian metric, Randers metric, Berwald metric, sectional curvature, flag curvature.