Lie Symmetries of the Canonical Connection: Codimension One Abelian Nilradical Case

This paper studies the canonical symmetric connection $\nabla$ associated to any Lie group $G$. The salient properties of $\nabla$ are stated and proved. The Lie symmetries of the geodesic system of a general linear connection are formulated. The results are then applied to $\nabla$ in the special case where the Lie algebra $\g$ of $G$, has a codimension one abelian nilradical. The conditions that determine a Lie symmetry in such a case are completely integrated. Finally the results obtained are compared with some four-dimensional Lie groups whose Lie algebras have three-dimensional abelian nilradicals, for which the calculations were performed by MAPLE.