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, ad ξ : N → Z {\text {ad}}\xi :\mathcal {N} \to \mathcal {Z} is surjective for all elements ξ ∈ N − Z \xi \in \mathcal {N} - \mathcal {Z} , where N \mathcal {N} denotes the Lie algebra of N and Z \mathcal {Z} denotes the center of N \mathcal {N} . Among other results we show that if H is a totally geodesic submanifold of N with dim ⁡ H ≥ 1 + dim ⁡ Z \dim H \geq 1 + \dim \mathcal {Z} , then H is an open subset of g N ∗ g{N^\ast } , where g is an element of H and N ∗ {N^\ast } is a totally geodesic subgroup of N . We find simple and useful criteria that are necessary and sufficient for a subalgebra N ∗ {\mathcal {N}^\ast } of N \mathcal {N} to be the Lie algebra of a totally geodesic subgroup N ∗ {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 .