On Degenerate 3-(α,δ)-Sasakian Manifolds

Abstract

We propose a new method to construct degenerate 3-(α, δ)-Sasakian manifolds as fiber products of Boothby-Wang bundles over hyperkähler manifolds. Subsequently, we study homogeneous degenerate 3-(α, δ)-Sasakian manifolds and prove that no non-trivial compact examples exist aswell as that there is exactly one family of nilpotent Lie groups with this geometry, the quaternionic Heisenberg groups.

A Bochner Technique For Foliations With Non-Negative Transverse Ricci Curvature

“…Therefore π : aut(M ) → iso(F), X → X ⊥ is a well-defined linear map. We can determine the kernel of π using an argument from [GRS,Lemma 4.2]:…”

“…In particular, if M arises via [GRS,Theorem 3.1] as a T 3 -bundle over a compact hyperkähler manifold N with integral Kähler classes, then dim Aut(M ) ≤ b 1 (N ) + 3. Remark 5.12.…”

“…If (M, F, g) are as in Theorem 1.7 and Ric T vanishes everywhere, then the dimension of the vector space iso(F) of transverse Killing vector fields of (F, g) is equal to b 1 (F). In particular, if M arises via [GRS,Theorem 3.1] as a T 3 -bundle over a compact hyperkähler manifold N with integral Kähler classes, then dim Aut(M ) ≤ b 1 (N ) + 3.…”