Communications in Analysis and Geometry
Volume 11 (2003)
Riemannian Submersions and Lattices in 2-step Nilpotent Lie Groups
Pages: 441 – 488
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: N → N. We show that every such N with a suitable left invariant metric is the base space of a Riemannian submersion and homomorphism 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 constructing a Lie algebra whose Killing form B is negative semidefinite. If B is negative definite, then we show that N* admits a (cocompact) lattice subgroup Γ*. Moreover 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 and homomorphism ρ : N* → N induces a Riemannian submersion ρ' : 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 prove the existence of lattices in simply connected, 2-step nilpotent Lie groups N that arise from Lie triple systems with compact center in so(n,ℝ), the Lie algebra of skew symmetric linear transformations of ℝn with the standard inner product. Lie triple systems with compact center include subspaces of so(n,ℝ) that arise from finite dimensional real representations of Clifford algebras or compact Lie groups. The center of the Lip triple system is trivial for representations of Clifford algebras and compact semisimple Lie groups.