On the Riemannian geometry of tangent Lie groups
Let \(G\) denote an \(m\)-dimensional Lie group. It is well known that if \(G\) is a Lie group, then its tangent bundle \(TG\) is also a Lie group. Moreover, the complete and vertical lifts of any left-invariant vector field of \(G\) are left-invariant vector fields on the Lie group \(TG\). In the present paper, the authors consider a Lie group \(G\) equipped with a left-invariant Riemannian metric \(g\). Using complete and vertical lifts of left-invariant vector fields, they induce a left-invariant Riemannian metric \(\widetilde{g}\) on the tangent Lie group \(TG\). The Levi-Civita connection and sectional curvature of \((TG,\overline{g})\) are given in terms of the Levi-Civita connection and the sectional curvature of \((G,g)\). Then, the formulas of Levi-Civita connection, sectional curvature and Ricci tensor of \((TG,\overline{g})\) are given in terms of structure constants of the Lie algebra of \(G\). These formulas are useful in constructing tangent Lie groups with some curvature properties. Finally, some examples of tangent Lie groups of strictly negative and non-negative Ricci curvatures are given.
scientific article; zbMATH DE number 6940995
scientific article