Left invariant lifted $(\alpha ,\beta )$-metrics of Douglas type on tangent Lie groups

Abstract

In this paper we study lifted left invariant $(\alpha ,\beta )$-metrics of Douglas type on tangent Lie groups. Let $G$ be a Lie group equipped with a left invariant $(\alpha ,\beta )$-metric of Douglas type $F$, induced by a left invariant Riemannian metric $g$. Using vertical and complete lifts, we construct the vertical and complete lifted $(\alpha ,\beta )$-metrics $F^v$ and $F^c$ on the tangent Lie group $TG$ and give necessary and sufficient conditions for them to be of Douglas type. Then, the flag curvature of these metrics are studied. Finally, as some special cases, the flag curvatures of $F^v$ and $F^c$ in the cases of Randers metrics of Douglas type, and Kropina and Matsumoto metrics of Berwald type are given.

References in corpus (5)

• Flag Curvature of Invariant Randers Metrics on Homogeneous Manifolds
• On the Randers metrics on two-step homogeneous nilmanifolds of dimension five
• Riemannian Geometry of Two Families of Tangent Lie Groups
• On the left invariant Randers and Matsumoto metrics of Berwald type on 3-dimensional Lie groups
• Left Invariant Randers Metrics of Berwald type on Tangent Lie Groups