On the twistor space of the six-sphere

Abstract

The set of all complex lines of the right-handed Dirac spinor bundle of a standard six-sphere is the total space of the twistor fibration. The twistor space, endowed with its natural Kähler structure, is recognised to be a six-dimensional complex quadric. The relevant group is Spin(7), which acts transitively on the six-quadric, as a group of fiber-preserving isometries. We use a result due to Berard-Bérgery and Matsuzawa to show the existence of a non-Kähler, non symmetric, Hermitian-Einstein metric on the six-quadric, which is Spin(7)-invariant.