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@article{1376701, author = {TaghaviandChabert, Arman}, article_location = {Amsterdam}, article_number = {April}, doi = {http://dx.doi.org/10.1016/j.difgeo.2017.02.008}, keywords = {Complex Riemannian geometry; Pure spinors; Distributions; Intrinsic torsion; Curvature prescription; Spinorial equations}, language = {eng}, issn = {0926-2245}, journal = {Differential Geometry and its Applications}, title = {Pure spinors, intrinsic torsion and curvature in odd dimensions}, volume = {51}, year = {2017} }
TY - JOUR ID - 1376701 AU - Taghavi-Chabert, Arman PY - 2017 TI - Pure spinors, intrinsic torsion and curvature in odd dimensions JF - Differential Geometry and its Applications VL - 51 IS - April SP - 117-152 EP - 117-152 PB - Elsevier Science SN - 09262245 KW - Complex Riemannian geometry KW - Pure spinors KW - Distributions KW - Intrinsic torsion KW - Curvature prescription KW - Spinorial equations N2 - We study the geometric properties of a $(2m + 1)$-dimensional complex manifold $M$ admitting a holomorphic reduction of the frame bundle to the structure group $P \subset Spin(2m + 1, C)$, 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 $N_\xi$ and of its rank-$(m + 1)$ orthogonal complement $N_\xi^\perp$ 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. Notably, we relate the integrability properties of $N_\xi$ and $N_\xi^\perp$ to the existence of solutions of odd- dimensional versions of the zero-rest-mass field equation. We give necessary and sufficient conditions for the almost null structure associated to a pure conformal Killing spinor to be integrable. Finally, we 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. ER -
TAGHAVI-CHABERT, Arman. Pure spinors, intrinsic torsion and curvature in odd dimensions. \textit{Differential Geometry and its Applications}. Amsterdam: Elsevier Science, 2017, vol.~51, April, p.~117-152. ISSN~0926-2245. Available from: https://dx.doi.org/10.1016/j.difgeo.2017.02.008.
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