Manifolds with Holonomy Sp(n)Sp(1)

Simon Salamon

This holonomy group from Berger's list characterizes the class of quaternion-kaehler (or "nearly hyperkaehler") manifolds of real dimension 4n. Each such manifold M carries a parallel 4-form, and is Einstein but is not in general Kaehler. In the case of positive scalar curvature, the twistor space (a 2-sphere bundle over M) is Kaehler, and it is an open question as to whether the only complete examples are homogeneous. This topic provides fascinating links between complex and Riemannian geometry that illustrate the power of spinor and twistor methods.

http://scgp.stonybrook.edu/video_portal/video.php?id=2783

Simons- Simons Collaboration on Special Holonomy Workshop