Geometry of left-invariant vector fields on Lie groups

TL;DR

This paper classifies and characterizes left-invariant vector fields on five-dimensional nilpotent Lie groups with Riemannian metrics, revealing their algebraic structure and properties such as being Killing or conformal.

Contribution

It provides a detailed algebraic classification of invariant vector fields on specific Lie groups, including their relation to the Lie algebra center and the nonexistence of certain fields.

Findings

Killing fields coincide with the Lie algebra center.

One-harmonic and conformal fields are necessarily Killing.

No nontrivial concurrent vector fields exist in this setting.

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.