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SUMMARY:CR-twistor spaces over manifolds with G_2- and Spin(7)-structures 
 - Hông Vân Lê (Czech Academy of Sciences)
DTSTART:20230222T160000Z
DTEND:20230222T170000Z
UID:TALK194308@talks.cam.ac.uk
CONTACT:Oscar Randal-Williams
DESCRIPTION:In 1984\, inspired by Penrose's successful twistor program\, L
 eBrun constructed a CR-twistor space over an arbitrary conformal Riemannia
 n  3-manifold and proved that the CR-structure is formally integrable. Thi
 s twistor construction has been generalized by Rossi in 1985 for m-dimensi
 onal Riemannian  manifolds endowed with a (m-1)fold vector cross product (
 VCP). In 2011 Verbitsky generalized LeBrun's construction of twistor-space
 s to 7-manifolds endowed with a G_2-structure.  In my talk I shall report 
 on my joint work with Domenico Fiorenza (arXiv:2203.04233) on a unificatio
 n and generalization of LeBrun's\, Rossi's and Verbitsky's construction to
  the case where a Riemannian  manifold (M\, g) has a VCP structure. I sha
 ll explain how to express the formal integrability of the CR-structure in 
 terms of a torsion tensor on   the twistor space\, which is a Grassmanian
  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 tenso
 r vanishes if and only if (M\, g) has constant curvature\, and the horizon
 tal component vanishes if and only if (M\,g) is a torsion-free G_2 or Spi
 n(7)-manifold. Finally I shall  discuss some open problems.
LOCATION:MR13
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