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

0572817 - MÚ 2024 RIV DE eng J - Článek v odborném periodiku
Fiorenza, D. - Le, Hong-Van
CR-twistor spaces over manifolds with G2 - and Spin(7)-structures.
Annali di Matematica Pura ed Applicata. Roč. 202, č. 4 (2023), s. 1931-1953. ISSN 0373-3114. E-ISSN 1618-1891
Institucionální podpora: RVO:67985840
Klíčová slova: formally integrable CR-structure * Frölicher-Nijenhuis bracket * invariant algebraic curvature
Obor OECD: Pure mathematics
Impakt faktor: 1, rok: 2022
Způsob publikování: Open access
https://doi.org/10.1007/s10231-023-01307-0

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.
Trvalý link: https://hdl.handle.net/11104/0343375