Weyl connections and the local sphere theorem for quaternionic contact structures |
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Authors: | Jesse Alt |
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Institution: | 1.School of Mathematics,University of the Witwatersrand,Johannesburg,South Africa |
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Abstract: | 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. |
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