On the Randers metrics on two-step homogeneous nilmanifolds of dimension five

TL;DR

This paper explores the geometry of five-dimensional two-step nilpotent Lie groups, deriving their curvature properties and identifying conditions for Randers metrics of Berwald type, revealing that such metrics exist only with a three-dimensional center.

Contribution

It provides explicit formulas for Levi-Civita connection, curvature, and Randers metrics on these groups, highlighting the unique conditions for Berwald type metrics.

Findings

Spaces have constant negative scalar curvature

Only groups with three-dimensional center admit Berwald type Randers metrics

Flag and sectional curvatures share the same sign

Abstract

In this paper we study the geometry of simply connected two-step nilpotent Lie groups of dimension five. We give the Levi-Civita connection, curvature tensor, sectional and scalar curvatures of these spaces and show that they have constant negative scalar curvature. Also we show that the only space which admits left invariant Randers metric of Berwald type has three dimensional center. In this case the explicit formula for computing flag curvature is obtained and it is shown that flag curvature and sectional curvature have the same sign.