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A Sasakian manifold is a contact manifold $(M,\theta)$ equipped with a special kind of Riemannian metric $g$, called a Sasakian metric.

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Quaternion-Sasakian manifolds and special holonomy Sasakian manifolds

Nearly-Kahler manifold are the 6-dimensional links of G2-cones, see https://en.wikipedia.org/wiki/Nearly_K%C3%A4hler_manifold I do not know about the others
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