Pure spinors, intrinsic torsion and curvature in even dimensions

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• 作者：Arman Taghavi-Chabert ^{taghavia@math.muni.cz" class="auth_mail" title="E-mail the corresponding author}
• 关键词：Complex Riemannian geometry ; Pure spinors ; Distributions ; Intrinsic torsion ; Curvature prescription ; Spinorial equations
• 刊名：Differential Geometry and its Applications
• 出版年：2016
• 出版时间：June 2016
• 年：2016
• 卷：46
• 期：Complete
• 页码：164-203
• 全文大小：826 K

文摘

We study the geometric properties of a 2m  -dimensional complex manifold M$\mathcal{M}$ admitting a holomorphic reduction of the frame bundle to the structure group P⊂Spin(2m,C)$P\subset \mathrm{Spin}(2m,\mathbb{C})$, the stabiliser of the line spanned by a pure spinor at a point. Geometrically, M$\mathcal{M}$ is endowed with a holomorphic metric g, a holomorphic volume form, a spin structure compatible with g, and a holomorphic pure spinor field ξ up to scale. The defining property of ξ is that it determines an almost null structure, i.e. an m  -plane distribution N_ξ${\mathcal{N}}_{\xi }$ along which g is totally degenerate.

We develop a spinor calculus, by means of which we encode the geometric properties of N_ξ${\mathcal{N}}_{\xi }$ corresponding to the algebraic properties of the intrinsic torsion of the P-structure. This is the failure of the Levi-Civita connection ∇ of g to be compatible with the P-structure. In a similar way, we examine the algebraic properties of the curvature of ∇.

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)$(\mathcal{M},g)$ has prescribed curvature.

We discuss applications of this work to the study of real pseudo-Riemannian manifolds.