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 )
https://doi.org/10.15407/mag17.02.201

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

Masumeh Nejadahm

Department of Mathematics, Isfahan University of Technology, Iran
E-mail: masumeh.nejadahmad@math.iut.ac.ir

Hamid Reza Salimi Moghaddam

Department of Pure Mathematics, Faculty of Mathematics and Statistics, University of Isfahan, Isfahan, 81746-73441, Iran
E-mail: hr.salimi@sci.ui.ac.ir, salimi.moghaddam@gmail.com

Received March 28, 2020, revised September 14, 2020.

Abstract

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.

Mathematics Subject Classification 2010: 53B21, 22E60, 22E15
Key words: left invariant (α, β)-metric, complete and vertical lifts, flag curvature

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