# Quaternion-Sasakian manifolds and special holonomy Sasakian manifolds

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?

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