Riemannian Submersions and Lattices in 2-step Nilpotent Lie Groups

We consider simply connected, 2-step nilpotent Lie groups N, all of which are diffeomorphic to Euclidean spaces via the Lie group exponential map exp : ˆ → N. We show that every such N with a suitable left invariant metric is the base space of a Riemannian submersion ρ : N* → N, where the fibers of ρ are flat, totally geodesic Euclidean spaces. The left invariant metric and Lie algebra of N* are obtained from N by a procedure involving the construction of a Lie algebra G whose Killing form B is negative semidefinite. If B is negative definite, then we show that N* admits a (cocompact) lattice subgroup Γ*, and Γ = ρ(Γ*) is a lattice in N if Γ* ∩ Ker(ρ) is a lattice in Ker(ρ). Conversely, if N admits a lattice Γ, then N* admits a lattice Γ* such that Γ = ρ(Γ*). In this case the Riemannian submersion ρ : N* → N induces a Riemannian submersion q : Γ* \ N* → Γ \ N whose fibers are flat, totally geodesic tori. The idea underlying the proof is that every 2-step nilpotent Lie algebra is isomorphic to a standard metric 2-step nilpotent Lie algebra, which we define and discuss. We also use a criterion of Mal'cev to derive conditions that guarantee the existence of lattices in N. We apply these conditions to three types of standard metric 2-step nilpotent Lie algebras that arise respectively from representations of Clifford algebras, representations of compact semisimple Lie groups and Lie triple systems in so(n,Â).