  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 Gau{\ss}-Bonnet theorem. We compute these invariants for the
Poincar\'e foliations on the 3-sphere and derive rigidity statements, including
a uniformisation theorem for orbifolds. These results are then applied to the
classification of taut contact circles.
