Pure spinors, intrinsic torsion and curvature in even dimensions

TAGHAVI-CHABERT, Arman. Pure spinors, intrinsic torsion and curvature in even dimensions. Online. Differential Geometry and its Applications. Elsevier Science, 2016, vol. 46, June, p. 164-203. ISSN 0926-2245. Available from: https://dx.doi.org/10.1016/j.difgeo.2016.02.006. [citováno 2024-04-24]

Basic information

• Original name: Pure spinors, intrinsic torsion and curvature in even dimensions
• Authors: TAGHAVI-CHABERT, Arman (250 France, guarantor, belonging to the institution)
• Edition: Differential Geometry and its Applications, Elsevier Science, 2016, 0926-2245.

Other information

• Original language: English
• Type of outcome: Article in a journal
• Field of Study: 10101 Pure mathematics
• Country of publisher: Netherlands
• Confidentiality degree: is not subject to a state or trade secret
• Impact factor: Impact factor: 0.497
• RIV identification code: RIV/00216224:14310/16:00088801
• Organization unit: Faculty of Science
• Doi: http://dx.doi.org/10.1016/j.difgeo.2016.02.006
• UT WoS: 000374599400010
• Keywords in English: Complex Riemannian geometry; Pure spinors Distributions; Intrinsic torsion; Curvature prescription; Spinorial equations
• Tags: AKR, rivok
• Tags: International impact, Reviewed

Abstract

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.

Links

• GP14-27885P, research and development project: Name: Skoro izotropní struktury v pseudo-riemannovské geometrii

Investor: Czech Science Foundation