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Left invariant lifted $(α,β)$-metrics of Douglas type on tangent Lie groups

Masumeh Nejadahmad, Hamid Reza Salimi Moghaddam

Abstract

In this paper we study lifted left invariant $(α,β)$-metrics of Douglas type on tangent Lie groups. Let $G$ be a Lie group equipped with a left invariant $(α,β)$-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 $(α,β)$-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.

Left invariant lifted $(α,β)$-metrics of Douglas type on tangent Lie groups

Abstract

In this paper we study lifted left invariant -metrics of Douglas type on tangent Lie groups. Let be a Lie group equipped with a left invariant -metric of Douglas type , induced by a left invariant Riemannian metric . Using vertical and complete lifts, we construct the vertical and complete lifted -metrics and on the tangent Lie group 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 and in the cases of Randers metrics of Douglas type, and Kropina and Matsumoto metrics of Berwald type are given.

Paper Structure

This paper contains 3 sections, 8 theorems, 28 equations.

Table of Contents

  1. Introduction
  2. Lifting of $(\alpha , \beta)$-metrics on Tangent Bundles
  3. Examples

Key Result

Theorem 1.1

Assume that $F=\alpha \phi(\frac{\beta}{\alpha})$ is a homogeneous $(\alpha , \beta)$-metric on $G/H$. Then $F$ is a Douglas metric if and only if either $F$ is a Berwald metric or $F$ is a Douglas metric of Randers type.

Theorems & Definitions (18)

  • Theorem 1.1
  • Lemma 2.1
  • proof
  • Lemma 2.2
  • proof
  • Remark 2.3
  • Lemma 2.4
  • proof
  • Theorem 2.5
  • proof
  • ...and 8 more