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Two well-known slogans are

A Sasakian manifold is the odd dimensional analogue of a Kähler manifold

and

A $3$-Sasakian manifold is the odd dimensional analogue of a hyper-Kähler manifold

Does this analogy extend to quaternion-Kähler manifolds? Is there a notion of quaternion-Sasakian manifolds?

What about special holonomy? Does there exist a notion of $Spin(7)$-Sasakian manifold. What about $G_2$-holonomy?

Can one use find an even dimensional analogue?

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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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  • $\begingroup$ There is a similar result between dimensions $7$ and $8$ too. $\endgroup$ – Paul Reynolds Apr 23 at 12:14

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