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Classification of Douglas $(α,β)$-metrics on five dimensional nilpotent Lie groups

Masoumeh Hosseini, Hamid Reza Salimi Moghaddam

Abstract

In this paper we classify all simply connected five dimensional nilpotent Lie groups which admit $(α,β)$-metrics of Berwald and Douglas type defined by a left invariant Riemannian metric and a left invariant vector field. During this classification we give the geodesic vectors, Levi-Civita connection, curvature tensor, sectional curvature and $S$-curvature.

Classification of Douglas $(α,β)$-metrics on five dimensional nilpotent Lie groups

Abstract

In this paper we classify all simply connected five dimensional nilpotent Lie groups which admit -metrics of Berwald and Douglas type defined by a left invariant Riemannian metric and a left invariant vector field. During this classification we give the geodesic vectors, Levi-Civita connection, curvature tensor, sectional curvature and -curvature.

Paper Structure

This paper contains 7 sections, 17 theorems, 20 equations.

Table of Contents

  1. Introduction
  2. Five dimensional two-step nilpotent Lie groups
  3. Five dimensional nilpotent Lie groups of nilpotency class greater than two
  4. Four-step nilpotent Lie groups with Lie algebra $\mathfrak{l}_{5,7}$
  5. Four-step nilpotent Lie groups with Lie algebra $\mathfrak{l}_{5,6}$
  6. Three-step nilpotent Lie groups with Lie algebra $\mathfrak{l}_{5,5}$
  7. Three-step nilpotent Lie groups with Lie algebra $\mathfrak{l}_{5,9}$

Key Result

Theorem 2.1

Let $G$ be a five dimensional two-step nilpotent Lie group equipped with a Randers metric of Douglas type arisen from a left invariant Riemannian metric $\tilde{a}$ and a left invariant vector field $X$. Then,

Theorems & Definitions (32)

  • Theorem 2.1
  • proof
  • Lemma 3.1
  • proof
  • Theorem 3.2
  • proof
  • Theorem 3.3
  • proof
  • Theorem 3.4
  • proof
  • ...and 22 more