Pure spinors, intrinsic torsion and curvature in odd dimensions

We develop a spinor calculus for a (2m+1)-dimensional complex Riemannian manifold (M,g) equipped with a preferred holomorphic projective pure spinor field [xi]. Such a spinor defines a holomorphic distribution N of m-planes, totally null with respect to g. The stabiliser of [xi] is a parabolic Lie subgroup P of SO(2m+1,\C), and using its algebraic properties, we give P-invariant classifications of the curvature tensors of the Levi-Civita connection, which, in the case of the Weyl tensor, generalises the Petrov-Penrose classification to odd dimensions. From a spinorial point of view, this generalises the notion of principal spinors. We also classify the intrinsic torsion of [xi] in terms of P-irreducibles, which gives an algebraic measure of the failure of [xi] to be parallel with respect to the Levi-Civita connection. In particular, we interpret the integrability properties of the associated null m-plane distribution N and its orthogonal complement N^perp in terms of the algebraic types of the intrinsic torsion of [xi]. The conformal invariance of these classes are studied and we give curvature conditions for a number of cases. We then investigate the geometric properties of a number of spinorial differential equations. Notably, we relate the integrability properties of a null $m$-plane distribution to the existence of solutions of odd-dimensional versions of the zero-rest-mass field equation. We also give necessary and sufficient conditions for a pure conformal Killing spinor to be foliating. This leads in particular to a conjecture refining the author's odd-dimensional generalisation of the Goldberg-Sachs theorem. This work can also be applied to a smooth real pseudo-Riemannian manifold of split signature equipped with a preferred projective real pure spinor field.