Invariant distributions and the transport twistor space of closed surfaces

TL;DR

This paper explores invariant distributions on the tangent bundle of closed surfaces, establishing their algebraic structure, linking them to holomorphic functions on twistor space, and classifying certain line bundles, with implications for tensor tomography.

Contribution

It introduces a new algebraic structure for invariant distributions, establishes a bijection with holomorphic functions on twistor space, and classifies holomorphic line bundles on twistor space for Anosov surfaces.

Findings

Invariant distributions form a unital algebra.

Bijection between invariant distributions and holomorphic functions on twistor space.

Classification of holomorphic line bundles on twistor space for Anosov surfaces.

Abstract

The purpose of this paper is to study transport equations on the unit tangent bundle of closed oriented Riemannian surfaces and to connect these to the transport twistor space of the surface (a complex surface naturally tailored to the geodesic vector field). We show that fibrewise holomorphic distributions invariant under the geodesic flow - which play an important role in tensor tomography on surfaces - form a unital algebra, that is, multiplication of such distributions is well-defined and continuous. We also exhibit a natural bijective correspondence between fibrewise holomorphic invariant distributions and genuine holomorphic functions on twistor space with polynomial blowup on the boundary of the twistor space. Eventually, when the surface is Anosov, we classify holomorphic line bundles over twistor space which are smooth up to the boundary. As a byproduct of our analysis, we obtain a quantitative version of a result of Flaminio, asserting that invariant distributions of the geodesic flow of a positively-curved metric on the 2-sphere are determined by their zeroth and first Fourier modes.