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

[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 $$N^*$$.

### MSC:

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

Zbl 0820.53047
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