Weyl connections and the local sphere theorem for quaternionic contact structures
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We apply the theory of Weyl structures for parabolic geometries developed by Čap and Slovák (Math Scand 93(1):53–90, 2003) to compute, for a quaternionic contact (qc) structure, the Weyl connection associated to a choice of scale, i.e. to a choice of Carnot–Carathéodory metric in the conformal class. The result of this computation has applications to the study of the conformal Fefferman space of a qc manifold, cf. (Geom Appl 28(4):376–394, 2010). In addition to this application, we are also able to easily compute a tensorial formula for the qc analog of the Weyl curvature tensor in conformal geometry and the Chern–Moser tensor in CR geometry. This tensor was first discovered via different methods by Ivanov and Vasillev (J Math Pures Appl 93:277–307, 2010), and we also get an independent proof of their Local Sphere Theorem. However, as a result of our derivation of this tensor, its fundamental properties—conformal covariance, and that its vanishing is a sharp obstruction to local flatness of the qc structure—follow as easy corollaries from the general parabolic theory.
KeywordsQuaternionic contact structures Weyl structures Parabolic geometry Equivalence problem
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- 2.Biquard, O.: Métriques d’Einstein asymptotiquement symétriques, Astérisque 265 (2000) (English translation: Asymptotically Symmetric Einstein Metrics, SMF/AMS Texts and Monographs, Vol. 13 105 +v pp. (English translation) (2006)).Google Scholar
- 7.Čap, A., Souček, V.: Subcomplexes in curved BGG-sequences, ESI preprint 1683Google Scholar
- 12.Ivanov, S., Minchev, I., Vassilev, D.: Quaternionic contact Einstein structures and the quaternionic contact Yamabe problem. arXiv preprint math.DG/0611658. ICTP Preprint: IC/2006/117Google Scholar
- 13.Landsberg, J.M., Robles, C.: Fubini-Griffiths-Harris rigidity and lie algebra cohomology. Preprint arXiv:0707.3410Google Scholar
- 15.Sharpe R.W.: Differential geometry. Springer, Berlin (1999)Google Scholar
- 16.Šilhan, J.: Algorithmic Computations of Lie Algebra Cohomologies. Supplemento ai Rendiconti del Circolo Matematico di Palermo, Serie II, Palermo, pp. 191–197 (2003)Google Scholar
- 18.Weyl, H.: Raum, Zeit, Materie. Springer, Berlin (1918) (English translation: Space, Time, Matter. Dover, New York (1952))Google Scholar
- 19.Yamaguchi K.: Differential systems associated with simple graded Lie algebras. Adv. Stud. Pure Math. 22, 413–494 (1993)Google Scholar