Geometry of left-invariant vector fields on Lie groups
Abstract
This paper examines the geometry of left-invariant vector fields on five-dimensional, simply connected, nilpotent Lie groups equipped with left-invariant Riemannian metrics. Using the canonical identification between the Lie algebra and the space of left-invariant vector fields, we derive algebraic characterizations for Killing, one-harmonic Killing, conformal, and concurrent vector fields. Employing a classification of these nilpotent Lie algebras into ten canonical types, we perform a case-by-case analysis to determine the structure of these vector fields. We prove that the space of Killing fields coincides with the center of the Lie algebra, and that one-harmonic and conformal fields are necessarily Killing. Furthermore, we establish the nonexistence of nontrivial concurrent vector fields in this setting.
Cite
@article{arxiv.2508.11440,
title = {Geometry of left-invariant vector fields on Lie groups},
author = {M. L. Foka and R. P. Nimpa and M. B. N. Djiadeu},
journal= {arXiv preprint arXiv:2508.11440},
year = {2025}
}