Geometry of left-invariant vector fields on Lie groups

Abstract

We investigate the geometry of left-invariant vector fields on simply connected nilpotent Lie groups equipped with left-invariant Riemannian metrics. Exploiting the canonical identification between the Lie algebra $\mathfrak{g}$ and the space of left-invariant vector fields, we establish complete algebraic characterizations for several fundamental classes of vector fields, including Killing, one-harmonic, harmonic, conformal, and concurrent fields. Our main results reveal a striking rigidity phenomenon: on any nilpotent Lie group, the spaces of Killing, one-harmonic, and conformal vector fields all coincide precisely with the center of the Lie algebra. Moreover, we prove that no nontrivial concurrent vector fields exist in this setting. In contrast, harmonic vector fields form a proper subspace of the Lie algebra, characterized as those central vectors orthogonal to the derived algebra.