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Non-Berwaldian Randers metrics of Douglas type on four-dimensional hypercomplex Lie groups

M. Hosseini, H. R. Salimi Moghaddam

Abstract

In this paper we classify all non-Berwaldian Randers metrics of Douglas type arising from invariant hyper-Hermitian metrics on simply connected four-dimensional real Lie groups. Also, the formulas of the flag curvature are given and it is shown that, in some directions, the flag curvature of the Randers metrics and the sectional curvature of the hyper-Hermitian metrics have the same sign.

Non-Berwaldian Randers metrics of Douglas type on four-dimensional hypercomplex Lie groups

Abstract

In this paper we classify all non-Berwaldian Randers metrics of Douglas type arising from invariant hyper-Hermitian metrics on simply connected four-dimensional real Lie groups. Also, the formulas of the flag curvature are given and it is shown that, in some directions, the flag curvature of the Randers metrics and the sectional curvature of the hyper-Hermitian metrics have the same sign.

Paper Structure

This paper contains 3 sections, 3 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Randers metrics of Douglas type and their flag curvature

Key Result

Theorem 3.1

Let $G$ be a non-commutative four-dimensional hypercomplex simply connected Lie group equipped with a left invariant hyper-Hermitian metric $g$. Suppose that $F$ is a non-Berwaldian Randers metric of Douglas type defined by $g$ and a left invariant vector field $Q$. Then the Lie algebra $\frak{g}$ o where $p, q\in \Bbb{R}$ and $\{ X, Y, Z, W\}$ is an orthonormal basis for $\mathfrak{g}$ with respe

Theorems & Definitions (5)

  • Theorem 3.1
  • proof
  • Corollary 3.2
  • Corollary 3.3
  • proof