Cohomology in algebra, geometry, physicsand statistics

CR-twistor spaces over manifolds with $G_2$ -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 product (VCP).  In 2011 Verbitsky   generalized    LeBrun's construction   of   twistor-spaces     to   $7$-manifolds  endowed  with    a $G_2$-structure.  In my talk I shall explain how to 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. Then  I shall show  that the  formal integrability of the CR-structure is expressed  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,  the vertical component of  the  torsion tensor  vanishes,  if and only if  $(M, g)$ has constant curvature,  and the horizontal component  vanishes,     if    $(M,g)$  is a  torsion-free $G_2$ or $Spin(7)$-manifold. Finally I shall discuss related open problems. This  is a joint  work with Domenico Fiorenza, https://arxiv.org/abs/2203.04233 .

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