Totally geodesic subalgebras in 2-step nilpotent Lie algebras (Q908250)

The authors describe totally geodesic subalgebras of a metric \(2\)-step nilpotent Lie algebra \(\mathfrak{n}\). It is shown that a totally geodesic subalgebra of \(\mathfrak{n}\) is either abelian and flat or can be decomposed as a direct sum determined by the curvature transformation. They also give conditions under which a totally geodesic submanifold of a simply connected \(2\)-step nilpotent Lie group is a totally geodesic subgroup. They follow \textit{P. Eberlein}'s paper [Trans. Am. Math. Soc. 343, No. 2, 805--828 (1994; Zbl 0830.53039)] in which the condition of nonsingularity is imposed on \(\mathfrak{n}\). They remove this restriction and illustrate the distinction between the nonsingular case and the unrestricted case.