In 1984 LeBrun constructed a CR-twistor space over an arbitrary conformal Riemannian 3-manifold and proved that the CR-structure is formally integrable. This twistor construction has been generalized by Rossi in 1985 for m-dimensional Riemannian manifolds endowed with a (m - 1)-fold vector cross product (VCP). In 2011 Verbitsky generalized LeBrun's construction of twistor-spaces to 7-manifolds endowed with a G(2)-structure. In this paper we unify and generalize LeBrun's, Rossi's and Verbitsky's construction of a CR-twistor space to the case where a Riemannian manifold (M , g) has a VCP structure. We show that the formal integrability of the CR-structure is expressed in terms of a torsion tensor on the twistor space, which is a Grassmannian bundle over (M , g). If the VCP structure on (M , g) is generated by a G(2)- or Spin(7)-structure, then the vertical component of the torsion tensor vanishes if and only if (M , g) has constant curvature, and the horizontal component vanishes if and only if (M , g) is a torsion-free G(2) or Spin(7)-manifold. Finally we discuss some open problems.
CR-twistor spaces over manifolds with $$G_2$$- and Spin(7)-structures / Fiorenza, Domenico; V??n L??, H??ng. - In: ANNALI DI MATEMATICA PURA ED APPLICATA. - ISSN 0373-3114. - 202:4(2023), pp. 1931-1953. [10.1007/s10231-023-01307-0]
CR-twistor spaces over manifolds with $$G_2$$- and Spin(7)-structures
Domenico Fiorenza;
2023
Abstract
In 1984 LeBrun constructed a CR-twistor space over an arbitrary conformal Riemannian 3-manifold and proved that the CR-structure is formally integrable. This twistor construction has been generalized by Rossi in 1985 for m-dimensional Riemannian manifolds endowed with a (m - 1)-fold vector cross product (VCP). In 2011 Verbitsky generalized LeBrun's construction of twistor-spaces to 7-manifolds endowed with a G(2)-structure. In this paper we unify and generalize LeBrun's, Rossi's and Verbitsky's construction of a CR-twistor space to the case where a Riemannian manifold (M , g) has a VCP structure. We show that the formal integrability of the CR-structure is expressed in terms of a torsion tensor on the twistor space, which is a Grassmannian bundle over (M , g). If the VCP structure on (M , g) is generated by a G(2)- or Spin(7)-structure, then the vertical component of the torsion tensor vanishes if and only if (M , g) has constant curvature, and the horizontal component vanishes if and only if (M , g) is a torsion-free G(2) or Spin(7)-manifold. Finally we discuss some open problems.File | Dimensione | Formato | |
---|---|---|---|
Fiorenza_CR-twistor-spaces_2023 .pdf
solo gestori archivio
Note: versione editoriale
Tipologia:
Versione editoriale (versione pubblicata con il layout dell'editore)
Licenza:
Tutti i diritti riservati (All rights reserved)
Dimensione
482.02 kB
Formato
Adobe PDF
|
482.02 kB | Adobe PDF | Contatta l'autore |
Fiorenza_preprint_CR-twistor-spaces_2023 .pdf
accesso aperto
Note: preprint
Tipologia:
Documento in Pre-print (manoscritto inviato all'editore, precedente alla peer review)
Licenza:
Tutti i diritti riservati (All rights reserved)
Dimensione
429.61 kB
Formato
Adobe PDF
|
429.61 kB | Adobe PDF |
I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.