Motivated by the moduli theory of taut contact circles on spherical 3-manifolds, we relate taut contact circles to transversely holomorphic flows. We give an elementary survey of such 1-dimensional foliations from a topological viewpoint. We describe a complex analogue of the classical Godbillon–Vey invariant, the so-called Bott invariant, and a logarithmic monodromy of closed leaves. The Bott invariant allows us to formulate a generalised Gauß–Bonnet theorem. We compute these invariants for the Poincaré foliations on the 3-sphere and derive rigidity statements, including a uniformisation theorem for orbifolds. These results are then applied to the classication of taut contact circles.
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Hansjörg Geiges, Jesús Gonzalo Pérez, Transversely holomorphic flows and contact circles on spherical 3-manifolds. Enseign. Math. 62 (2016), no. 3/4, pp. 527–567DOI 10.4171/LEM/62-3/4-8