Riemannian geometry of tangent Lie groups using two left invariant Riemannian metrics

Abstract

In this paper, we consider a Lie group $G$ equipped with two left-invariant Riemannian metrics $g^1$ and $g^2$. Using these two left-invariant Riemannian metrics we define a left-invariant Riemannian metric $\tilde{g}$ on the tangent Lie group $TG$. The Levi-Civita connection, tensor curvature, and sectional curvature of $(TG,\tilde{g})$ in terms of $g^1$ and $g^2$ are given. Also, we give a sufficient condition for $\tilde{g}$ to be bi-invariant. Finally, motivated by the recent work of D. N. Pham, using symplectic forms ${\omega }_1$ and ${\omega }_2$ on $G$ we define a symplectic form $\tilde{\omega }$ on $TG$.