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?

  • $\begingroup$ There is a torsion free connection. But the story is complicated. $\endgroup$ – Ben McKay Apr 12 at 18:50
  • $\begingroup$ It is different from the Levi--Civita connection? $\endgroup$ – Boris Henriques Apr 12 at 18:53
  • $\begingroup$ It is the Levi-Civita. You can read about this stuff in Dominic Joyce's book on holonomy groups, I think, and in papers of Robert Bryant. $\endgroup$ – Ben McKay Apr 12 at 19:20
  • $\begingroup$ But I guess there is a different construction, analogous to the construction of the Chern for complex manifolds? $\endgroup$ – Boris Henriques Apr 12 at 19:26
  • $\begingroup$ I don't know of a different construction. You could say that you start with a connection, torsion-free, and ask what holonomy it could have. If it has holonomy group Sp(1)Sp(n), there is a unique parallel metric tensor, and various other parallel tensors, to make a quaternionic Kaehler manifold. $\endgroup$ – Ben McKay Apr 12 at 20:28

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