Classification of Douglas $(\alpha,\beta)$-metrics on five dimensional nilpotent Lie groups

TL;DR

This paper classifies five-dimensional nilpotent Lie groups that admit specific Finsler metrics, providing detailed geometric properties and curvature calculations for these groups.

Contribution

It offers a complete classification of five-dimensional nilpotent Lie groups with $(eta,eta)$-metrics of Berwald and Douglas type, including explicit geometric data.

Findings

Classification of all such Lie groups.

Explicit formulas for geodesic vectors and curvature.

Detailed geometric properties of the classified groups.

Abstract

In this paper we classify all simply connected five dimensional nilpotent Lie groups which admit $(\alpha,\beta)$-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.