On the Riemannian geometry of tangent Lie groups

Abstract

In the present article we consider a Lie group G equipped with a left invariant Riemannian metric g. Then, by using complete and vertical lifts of left invariant vector fields we induce a left invariant Riemannian metric \(\widetilde{g}\) on the tangent Lie group TG. The Levi-Civita connection and sectional curvature of \((TG,\widetilde{g})\) are given, in terms of Levi-Civita connection and sectional curvature of (G, g). Then, we present Levi-Civita connection, sectional curvature and Ricci tensor formulas of \((TG,\widetilde{g})\) in terms of structure constants of the Lie algebra of G. Finally, some examples of tangent Lie groups of strictly negative and non-negative Ricci curvatures are given.