Geometry of $2$-step nilpotent groups with a left invariant metric. II

Abstract:

We obtain a partial description of the totally geodesic submanifolds of a 2-step, simply connected nilpotent Lie group with a left invariant metric. We consider only the case 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 $\mathcal {N}$. Among other results we show 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 $g{N^\ast }$, where g is an element of H and ${N^\ast }$ is a totally geodesic subgroup of N. We find simple and useful criteria that are necessary and sufficient for a subalgebra ${\mathcal {N}^\ast }$ of $\mathcal {N}$ to be the Lie algebra of a totally geodesic subgroup ${N^\ast }$. We define and study the properties of a Gauss map of a totally geodesic submanifold H of N. We conclude with a characterization of 2-step nilpotent Lie groups N of Heisenberg type in terms of the abundance of totally geodesic submanifolds of N.