CR-twistor spaces over manifolds with G2- and Spin(7)-structures

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...

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Bibliographic Details
Published in:Annali di matematica pura ed applicata 2023, Vol.202 (4), p.1931-1953
Main Authors: Fiorenza, Domenico, Lê, Hông Vân
Format: Article
Language:English
Subjects:
Mathematical analysis
Mathematics
Mathematics and Statistics
Riemann manifold
Tensors
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Summary: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.
ISSN:0373-3114
1618-1891
DOI:10.1007/s10231-023-01307-0