arXiv subjects
Publications and source records attributed to Leon Roschig.
We generalize the Bochner technique to foliations with non-negative transverse Ricci curvature. In particular, we obtain a new vanishing theorem for basic cohomology. Subsequently, we provide two natural applications, namely to degenerate 3-$(α,δ) $-Sasaki and certain Sasaki-$ η$-Einstein manifolds, which arise for example as Boothby-Wang bundles over hyperkähler and Calabi-Yau manifolds, respectively.
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 as well as that there is exactly one family of nilpotent Lie groups with this geometry, the quaternionic Heisenberg groups.
We provide a new, self-contained proof of the classification of homogeneous 3-Sasakian manifolds, which was originally obtained by Boyer, Galicki and Mann. In doing so, we construct an explicit one-to-one correspondence between simply connected homogeneous 3-Sasakian manifolds and simple complex Lie algebras via the theory of root systems. We also discuss why the real projective spaces are the only non-simply connected homogeneous 3-Sasakian manifolds and derive the famous classification of homogeneous positive quaternionic Kähler manifolds due to Wolf and Alekseevskii from our results.