Pure spinors, intrinsic torsion and curvature in even dimensions
| Authors | |
|---|---|
| Year of publication | 2016 |
| Type | Article in Periodical |
| Magazine / Source | Differential Geometry and its Applications |
| MU Faculty or unit | |
| Citation | |
| Doi | http://dx.doi.org/10.1016/j.difgeo.2016.02.006 |
| Field | General mathematics |
| Keywords | Complex Riemannian geometry; Pure spinors Distributions; Intrinsic torsion; Curvature prescription; Spinorial equations |
| Description | We study the geometric properties of a $2m$-dimensional complex manifold $M$ admitting a holomorphic reduction of the frame bundle to the structure group $P \subset Spin(2m, C)$, the stabiliser of the line spanned by a pure spinor at a point. Geometrically, $M$ is endowed with a holomorphic metric $g$, a holomorphic volume form, a spin structure compatible with $g$, and a holomorphic pure spinor field $\xi$ up to scale. The defining property of $\xi$ is that it determines an almost null structure, i.e. an $m$-plane distribution $N_\xi$ along which $g$ is totally degenerate. We develop a spinor calculus, by means of which we encode the geometric properties of $N_\xi$ corresponding to the algebraic properties of the intrinsic torsion of the $P$-structure. This is the failure of the Levi-Civita connection $\nabla$ of $g$ to be compatible with the $P$ -structure. In a similar way, we examine the algebraic properties of the curvature of $\nabla$. Applications to spinorial differential equations are given. In particular, we give necessary and sufficient conditions for the almost null structure associated to a pure conformal Killing spinor to be integrable. We also conjecture a Goldberg–Sachs-type theorem on the existence of a certain class of almost null structures when $(M, g)$ has prescribed curvature. We discuss applications of this work to the study of real pseudo-Riemannian manifolds. |
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