9 - Higher-Dimensional Descriptions

Summary

This Chapter describes the geometry of twistor space of a 4-dimensional manifold. We motivated the twistor space as a geometrical construction that realisesthe action of the conformal group in 4D as the direct analog of that in 2D. This explains why the coordinates of a 4D space can be naturally put into a 2x2 matrix. We describe both the complexified version of the twistor space, as well as treat all the 3 possible signatures in detail. We then specialise to the case of Euclidean twistors, and describe how the twistor space can be interpreted as the total space of the bundle of almost complex structures of a 4D Riemannian manifold. Quaternionic Hopf fibration and its relation to the Euclidean twistor space is desccribed. We then describe the geometry of 3-forms in seven dimensions, and describe two different G2 structures on the 7-sphere. We end with a description of a lift of the usual twistor construction of integrable almost complex structures into seven dimensions. This is based on the notion of nearly parallel G2 structures.