A Projective-to-Conformal Fefferman-Type Construction

J 2017

A Projective-to-Conformal Fefferman-Type Construction

HAMMERL, Matthias, Katja SAGERSCHNIG, Josef ŠILHAN, Arman TAGHAVI-CHABERT, Vojtěch ŽÁDNÍK et. al.

Basic information

Original name

A Projective-to-Conformal Fefferman-Type Construction

Authors

HAMMERL, Matthias (40 Austria), Katja SAGERSCHNIG (40 Austria), Josef ŠILHAN (203 Czech Republic, belonging to the institution), Arman TAGHAVI-CHABERT (250 France, belonging to the institution) and Vojtěch ŽÁDNÍK (203 Czech Republic, belonging to the institution)

Edition

Symmetry, Integrability and Geometry: Methods and Applications (SIGMA), 2017, 1815-0659

Other information

Language

English

Type of outcome

Článek v odborném periodiku

Field of Study

10101 Pure mathematics

Country of publisher

Ukraine

Confidentiality degree

není předmětem státního či obchodního tajemství

References:

URL

Impact factor

Impact factor: 1.100

RIV identification code

RIV/00216224:14310/17:00095269

Organization unit

Faculty of Science

DOI

http://dx.doi.org/10.3842/SIGMA.2017.081

UT WoS

000414168700001

Keywords in English

parabolic geometry; projective structure; conformal structure; Cartan connection; Fefferman spaces; twistor spinors

Abstract

V originále

We study a Fefferman-type construction based on the inclusion of Lie groups SL(n + 1) into Spin(n + 1, n + 1). The construction associates a split-signature (n, n)-conformal spin structure to a projective structure of dimension n. We prove the existence of a canonical pure twistor spinor and a light-like conformal Killing field on the constructed conformal space. We obtain a complete characterisation of the constructed conformal spaces in terms of these solutions to overdetermined equations and an integrability condition on the Weyl curvature. The Fefferman-type construction presented here can be understood as an alternative approach to study a conformal version of classical Patterson-Walker metrics as discussed in recent works by Dunajski-Tod and by the authors. The present work therefore gives a complete exposition of conformal Patterson-Walker metrics from the viewpoint of parabolic geometry.

Links

GA201/08/0397, research and development project

Name: Algebraické metody v geometrii a topologii

Investor: Czech Science Foundation, Algebraic methods in geometry and topology

GBP201/12/G028, research and development project

Name: Ústav Eduarda Čecha pro algebru, geometrii a matematickou fyziku

Investor: Czech Science Foundation

GP14-27885P, research and development project

Name: Skoro izotropní struktury v pseudo-riemannovské geometrii

Investor: Czech Science Foundation

Citovat

HAMMERL, Matthias, Katja SAGERSCHNIG, Josef ŠILHAN, Arman TAGHAVI-CHABERT and Vojtěch ŽÁDNÍK. A Projective-to-Conformal Fefferman-Type Construction. Symmetry, Integrability and Geometry: Methods and Applications (SIGMA). 2017, vol. 13, No 81, p. 1-33. ISSN 1815-0659. Available from: https://dx.doi.org/10.3842/SIGMA.2017.081.

@article{1399439,
   author = {Hammerl, Matthias and Sagerschnig, Katja and Šilhan, Josef and TaghaviandChabert, Arman and Žádník, Vojtěch},
   article_number = {81},
   doi = {http://dx.doi.org/10.3842/SIGMA.2017.081},
   keywords = {parabolic geometry; projective structure; conformal structure; Cartan connection; Fefferman spaces; twistor spinors},
   language = {eng},
   issn = {1815-0659},
   journal = {Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)},
   title = {A Projective-to-Conformal Fefferman-Type Construction},
   url = {https://www.emis.de/journals/SIGMA/2017/081/},
   volume = {13},
   year = {2017}
}

TY  - JOUR
ID  - 1399439
AU  - Hammerl, Matthias - Sagerschnig, Katja - Šilhan, Josef - Taghavi-Chabert, Arman - Žádník, Vojtěch
PY  - 2017
TI  - A Projective-to-Conformal Fefferman-Type Construction
JF  - Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)
VL  - 13
IS  - 81
SP  - 1-33
EP  - 1-33
SN  - 18150659
KW  - parabolic geometry
KW  - projective structure
KW  - conformal structure
KW  - Cartan connection
KW  - Fefferman spaces
KW  - twistor spinors
UR  - https://www.emis.de/journals/SIGMA/2017/081/
L2  - https://www.emis.de/journals/SIGMA/2017/081/
N2  - We study a Fefferman-type construction based on the inclusion of Lie groups SL(n + 1) into Spin(n + 1, n + 1). The construction associates a split-signature (n, n)-conformal spin structure to a projective structure of dimension n. We prove the existence of a canonical pure twistor spinor and a light-like conformal Killing field on the constructed conformal space. We obtain a complete characterisation of the constructed conformal spaces in terms of these solutions to overdetermined equations and an integrability condition on the Weyl curvature. The Fefferman-type construction presented here can be understood as an alternative approach to study a conformal version of classical Patterson-Walker metrics as discussed in recent works by Dunajski-Tod and by the authors. The present work therefore gives a complete exposition of conformal Patterson-Walker metrics from the viewpoint of parabolic geometry.
ER  -

HAMMERL, Matthias, Katja SAGERSCHNIG, Josef ŠILHAN, Arman TAGHAVI-CHABERT and Vojtěch ŽÁDNÍK. A Projective-to-Conformal Fefferman-Type Construction. \textit{Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)}. 2017, vol.~13, No~81, p.~1-33. ISSN~1815-0659. Available from: https://dx.doi.org/10.3842/SIGMA.2017.081.

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