For a Riemannian manifold, the natural connection is of course the Levi-Civita connection. For a complex manifold, the natural connection is the Chern connection, which coincides with the Levi-Civita when the manifold is Kähler.
Question. What happens in the quaternionic Kähler case? Is there a "quaternionic Chern connection" for hyper-Kähler manifolds and, if so, does it coincide with the Levi-Civita connection? On another note, what happens for manifolds of exceptional holonomy?