Affine and Projective vector fields on five-dimensional nilpotent Lie groups

Abstract

This paper presents a complete classification of left-invariant affine and projective vector fields on five-dimensional simply connected nilpotent Lie groups endowed with Riemannian metrics. Building on the classification of left-invariant metrics on five-dimensional nilpotent Lie groups made by FOKa et al., we develop an algebraic characterization of these vector fields on an arbitrary Riemannian Lie group. We then employ this framework to classify left-invariant affine and projective vector fields on simply connected, Riemannian nilpotent Lie groups of dimension five. Key results demonstrate that all projective vector fields in this context are necessarily affine, extending classical results by [Kobayashi1963] on homogeneous spaces. We provide explicit matrix representations of the relevant operators and solve the resulting systems of equations case-by-case using algebraic techniques.