We show that every 3-\((\alpha ,\delta )\)-Sasaki manifold of dimension \(4n + 3\) admits a locally defined Riemannian submersion over a quaternionic Kähler manifold of scalar curvature \(16n(n+2)\alpha \delta\). In the non-degenerate case we describe all homogeneous 3-\((\alpha ,\delta )\)-Sasaki manifolds fibering over symmetric Wolf spaces and over their non-compact dual symmetric spaces. 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 non-symmetric Alekseevsky spaces, the lowest possible dimension of such a manifold being 19.

我们证明，每3维\（（\ alpha，\ delta）\）- Sasaki流形的尺寸\（4n + 3 \）都允许在标量曲率\（16n（n + 2） ）\ alpha \ delta \）。在非简并的情况下，我们描述了在对称Wolf空间及其非紧对对称空间上的所有齐次3- \（（\ alpha，\ delta）\）- Sasaki流形。如果\（\ alpha \ delta> 0 \），则会产生齐次3- \（（\ alpha，\ delta）\）- Sasaki流形的完整分类。对于\（\ alpha \ delta <0 \），我们提供了均质3- \（（\ alpha，\ delta）\）的一般构造-在不对称的Alekseevsky空间上形成纤维的Sasaki流形，这种流形的最小尺寸为19。