Homogeneous non-degenerate $3$-$(\alpha,\delta)$-Sasaki manifolds and submersions over quaternionic K\"ahler spaces

TL;DR

This paper classifies homogeneous 3-$(eta, au)$-Sasaki manifolds, showing they admit Riemannian submersions over quaternionic K"ahler spaces, and provides explicit constructions and classifications based on the parameters' signs.

Contribution

It offers a complete classification of homogeneous 3-$(eta, au)$-Sasaki manifolds, detailing their fibering over symmetric and nonsymmetric quaternionic K"ahler spaces.

Findings

Every 3-$(eta, au)$-Sasaki manifold admits a Riemannian submersion over a quaternionic K"ahler manifold.

Homogeneous 3-$(eta, au)$-Sasaki manifolds fibering over symmetric Wolf spaces are classified.

Construction of homogeneous 3-$(eta, au)$-Sasaki manifolds over nonsymmetric Alekseevsky spaces is provided.

Abstract

We show that every $3$-$(\alpha,\delta)$-Sasaki manifold of dimension $4n + 3$ admits a locally defined Riemannian submersion over a quaternionic K\"ahler manifold of scalar curvature $16n(n+2)\alpha\delta$. In the non-degenerate case ($\delta\neq 0$) we describe all homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds fibering over symmetric Wolf spaces (case $\alpha\delta> 0$) and over their the noncompact dual symmetric spaces (case $\alpha\delta< 0$). If $\alpha\delta> 0$, this yields a complete classification of homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds; for $\alpha\delta< 0$, we provide a general construction of homogeneous $3$-$(\alpha,\delta)$-Sasaki manifolds fibering over nonsymmetric Alekseevsky spaces, the lowest possible dimension of such a manifold being $19$.