On Degenerate $3$-$(\alpha,\delta)$-Sasakian Manifolds

TL;DR

This paper introduces a new construction method for degenerate 3-$(l,d)$-Sasakian manifolds using fiber products of Boothby-Wang bundles over hyperka4hler manifolds, and classifies homogeneous cases, showing only trivial compact examples and the quaternionic Heisenberg groups as non-trivial nilpotent cases.

Contribution

It presents a novel construction approach for degenerate 3-$(l,d)$-Sasakian manifolds and classifies homogeneous instances, identifying the quaternionic Heisenberg groups as unique nilpotent examples.

Findings

No non-trivial compact homogeneous degenerate 3-$(l,d)$-Sasakian manifolds exist.

The quaternionic Heisenberg groups are the only nilpotent Lie groups with this geometry.

A new fiber product construction method for these manifolds is proposed.

Abstract

We propose a new method to construct degenerate $3$-$(\alpha,\delta)$-Sasakian manifolds as fiber products of Boothby-Wang bundles over hyperk\"ahler manifolds. Subsequently, we study homogeneous degenerate $3$-$(\alpha, \delta)$-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.