Journal of Mathematical Physics, Analysis, Geometry 2021, vol. 17, No 2, pp. 201-215   https://doi.org/10.15407/mag17.02.201     ( to contents , go back )

### Left Invariant Lifted (α, β)-metrics of Douglas Type on Tangent Lie Groups

Department of Mathematics, Isfahan University of Technology, Iran

In the paper, lifted left invariant $(\alpha,\beta)$-metrics of Douglas type on tangent Lie groups are studied. Suppose that $g$ is a left invariant Riemannian metric on a Lie group $G$, and $F$ is a left invariant $(\alpha,\beta)$-metric of Douglas type induced by $g$. Using vertical and complete lifts, we construct the vertical and complete lifted $(\alpha,\beta)$-metrics $F^v$ and $F^c$ on the tangent bundle $TG$ 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 $F^c$ are given for Randers metrics of Douglas type and Kropina and Matsumoto metrics of Berwald type.