Geodesic
On the geometry of some para-hypercomplex Lie groups
Archivum mathematicum, Tome 45 (2009) no. 3, pp. 159-170.

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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.
Classification : 53B35, 53C15, 53C60, 58B20
Keywords: para-hypercomplex structure; left invariant Riemannian metric; Randers metric; Berwald metric; sectional curvature; flag curvature 
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     author = {Salimi Moghaddam, H. R.},
     title = {On the geometry of some para-hypercomplex {Lie} groups},
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     zbl = {1212.53053},
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Salimi Moghaddam, H. R. On the geometry of some para-hypercomplex Lie groups. Archivum mathematicum, Tome 45 (2009) no. 3, pp. 159-170. http://geodesic.mathdoc.fr/item/ARM_2009__45_3_a0/