On the geometry of some para-hypercomplex Lie groups. (English) Zbl 1212.53053

Moghaddam, H. R. Salimi

Arch. Math., Brno 45, No. 3, 159-170 (2009).

Summary: Firstly, we study some left invariant Riemannian metrics on para-hypercomplex 4-dimensional Lie groups. For each Lie group, the Levi-Civita connection and sectional curvature are 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 the flag curvature are obtained in all cases. Some of these Finsler Lie groups are of non-positive flag curvature.

MSC:

53C15	General geometric structures on manifolds (almost complex, almost product structures, etc.)
58B20	Riemannian, Finsler and other geometric structures on infinite-dimensional manifolds
53B35	Local differential geometry of Hermitian and Kählerian structures
53C60	Global differential geometry of Finsler spaces and generalizations (areal metrics)

Keywords:

para-hypercomplex structure; left invariant Riemannian metric; Randers metric; Berwald metric; sectional curvature; flag curvature