Totally geodesic subalgebras in 2-step nilpotent Lie algebras

Abstract

We describe totally geodesic subalgebras of a metric 2-step nilpotent Lie algebra $\n$. We prove that a totally geodesic subalgebra of $\n$ is either abelian and flat or can be decomposed as a direct sum determined by the curvature transformation. In addition, we give conditions under which a totally geodesic submanifold of a simply connected 2-step nilpotent Lie group is a totally geodesic subgroup. We follow Eberlein's 1994 paper in which he imposes the condition of nonsingularity on $\n$. We remove this restriction and illustrate the distinction between the nonsingular case and the unrestricted case.