Transversely holomorphic flows and contact circles on spherical 3-manifolds

TL;DR

This paper explores the relationship between taut contact circles and transversely holomorphic flows on spherical 3-manifolds, introducing invariants and applying them to classify these structures.

Contribution

It introduces the Bott invariant as a complex analogue of Godbillon-Vey, and applies it to classify taut contact circles on spherical 3-manifolds.

Findings

Computed Bott invariants for Poincaré foliations on the 3-sphere.

Derived rigidity and uniformisation theorems for orbifolds.

Provided a topological survey of transversely holomorphic flows.

Abstract

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.