Twistor CR Manifolds by non-Riemannian ConnectionsLow, Ho Chi (2020) Twistor CR Manifolds by non-Riemannian Connections. Doctoral Dissertation, University of Pittsburgh. (Unpublished)
AbstractDeveloped by LeBrun, twistor CR manifold is a 5-dimensional CR manifold foliated by Riemann spheres. The CR structure is determined by both the complex structure on the Riemann sphere and the geometric information of the space of leaves, which is a 3-manifold endowed with a conformal class of metrics and a trace-free symmetric (1,1)-tensor. When the (1,1)-tensor is zero, the twistor CR structure of zero torsion, named as the rival CR structure on LeBrun's paper “Foliated CR Manifolds”, is obtained. These CR structures are embeddable to a complex 3-manifold if and only if the metric tensor is conformal to a real analytic metric. We try to understand twistor CR structures through the corresponding Fefferman metric defined on the canonical circle bundle of the given CR manifold. The conformal class of the Fefferman metric is preserved over the choice of contact forms of the CR structure, so it makes possible to classify CR structures by the confomal curvature tensor of the Fefferman metric. Our main results include representing the Weyl tensor in terms of the Cotton tensor on the 3-manifold when the twistor CR structure is of zero torsion. Moreover, we obtain conditions for vanishing Weyl tensor when the space of leaves is under a flat metric. Share
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