Geometry of 2-step nilpotent groups with a left invariant metric. II. (English) Zbl 0830.53039

Eberlein, Patrick

Trans. Am. Math. Soc. 343, No. 2, 805-828 (1994).

[For part I, cf. Ann. Sci. Éc. Norm. Supér., IV. Sér. 27, No. 5, 611-660 (1994; Zbl 0820.53047.]
In this paper, the author discusses the totally geodesic submanifolds of a 2-step, simply connected nilpotent Lie group \(N\) with a left invariant metric. The author assumes that \(N\) is nonsingular, that is, \(\text{ad }\xi : {\mathcal N} \to {\mathcal Z}\) is surjective for all elements \(\xi \in {\mathcal N} - {\mathcal Z}\), where \(\mathcal N\) denotes the Lie algebra of \(N\) and \(\mathcal Z\) denotes the center of \(\mathcal N\). The author proves that if \(H\) is a totally geodesic submanifold of \(N\) with \(\dim H \geq 1 + \dim {\mathcal Z}\), then \(H\) is an open subset of \(gN^*\), where \(g \in H\) and \(N^*\) is a totally geodesic subgroup of \(N\). And he finds a necessary and sufficient condition for a subalgebra \({\mathcal N}^*\) of \(\mathcal N\) to be the Lie algebra of a totally geodesic subgroup \(N^*\).

Reviewer: Hou Zixin (Tianjin)

MSC:

53C30	Differential geometry of homogeneous manifolds
22E25	Nilpotent and solvable Lie groups

Keywords:

2-step nilpotent Lie group; totally geodesic submanifold

Citations:

Zbl 0820.53047