Curvatures of left invariant Randers metric on the five-dimensional Heisenberg group

TL;DR

This paper investigates the curvature properties of left invariant Z-Randers metrics on the five-dimensional Heisenberg group, showing the existence of both strictly negative and positive curvature flags.

Contribution

It proves that all such metrics admit flags with strictly negative and positive curvatures, revealing diverse curvature behaviors within this class.

Findings

Existence of flags with strictly negative curvature

Existence of flags with strictly positive curvature

Curvature diversity in Z-Randers metrics on the Heisenberg group

Abstract

A left invariant Z-Randers metric on the five-dimensional Heisenberg group is a left invariant Randers metric with deformation vector from the center of the Heisenberg algebra. In this note we prove that for every left invariant Z-Randers metric on the five-dimensional Heisenberg group there exist flags of strictly negative and there exist flags of strictly positive curvatures.