nLab Wolf space

Idea

A space which is both a quaternion-Kähler manifold as well as a symmetric space. Also known as a Wolf space.

Examples

• quaternionic projective plane

Properties

As positive quaternion-Kähler manifolds

under construction

Proposition

Every Wolf space is a positive quaternion-Kähler manifold.

In fact the Wolf spaces are the only known examples of positive quaternion-Kähler manifold (which is not hyper-Kähler ?!), as of today (e.g. Salamon 82, Section 5).

This leads to the conjecture that un every dimension, the Wolf spaces are the only positive quaternion-Kähler manifolds.

The conjecture has been proven for the following dimensions

• $d = 4$ (Hitchin)

• $d = 8$ (Poon-Salamon 91, LeBrun-Salamon 94)

References

• Joseph K. Wolf, Complex homogeneous contact manifolds and quaternionic symmetric spaces, Journal of Math. and Mech., 14 (1965), p. 166 (jstor:24901319)

• Simon Salamon, Quaternionic Kähler manifolds, Invent Math (1982) 67: 143. (doi:10.1007/BF01393378)

• Y. S. Poon, Simon Salamon, Quaternionic Kähler 8-manifolds with positive scalar curvature, J. Differential Geom. Volume 33, Number 2 (1991), 363-378 (euclid:1214446322)

• Claude LeBrun, Simon Salamon, Strong rigidity of positive quaternion Kähler manifolds, Inventiones Mathematicae 118, 1994, 109–132 (dml:144231, doi:10.1007/BF01231528)

• Amann, Positive Quaternion Kähler Manifolds, 2009 (pdf)

See also

• Wikipedia, Quaternion-Kähler symmetric space