Pure spinors, intrinsic torsion and curvature in even
dimensions

J 2016

Pure spinors, intrinsic torsion and curvature in even dimensions

TAGHAVI-CHABERT, Arman

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

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

Netherlands

Confidentiality degree

není předmětem státního či obchodního tajemství

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

Abstract

V originále

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

Citovat

TAGHAVI-CHABERT, Arman. Pure spinors, intrinsic torsion and curvature in even dimensions. 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.

@article{1376706,
   author = {TaghaviandChabert, Arman},
   article_number = {June},
   doi = {http://dx.doi.org/10.1016/j.difgeo.2016.02.006},
   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 even dimensions},
   volume = {46},
   year = {2016}
}

TY  - JOUR
ID  - 1376706
AU  - Taghavi-Chabert, Arman
PY  - 2016
TI  - Pure spinors, intrinsic torsion and curvature in even dimensions
JF  - Differential Geometry and its Applications
VL  - 46
IS  - June
SP  - 164-203
EP  - 164-203
PB  - Elsevier Science
SN  - 09262245
KW  - Complex Riemannian geometry
KW  - Pure spinors Distributions
KW  - Intrinsic torsion
KW  - Curvature prescription
KW  - Spinorial equations
N2  - 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.
ER  -

TAGHAVI-CHABERT, Arman. Pure spinors, intrinsic torsion and curvature in even dimensions. \textit{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.

Detailed Information on Publication Record