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

## TL;DR

This paper investigates lifted left invariant $(eta,eta)$-metrics of Douglas type on tangent Lie groups, establishing conditions for their Douglas property and analyzing their flag curvature, including special cases like Randers, Kropina, and Matsumoto metrics.

## Contribution

It introduces necessary and sufficient conditions for lifted $(eta,eta)$-metrics to be of Douglas type on tangent Lie groups and studies their curvature properties.

## Key findings

- Conditions for lifted metrics to be of Douglas type
- Explicit formulas for flag curvature of lifted metrics
- Analysis of special cases like Randers, Kropina, and Matsumoto metrics

## 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.