Transparent pairs

TL;DR

This paper classifies cohomologically trivial SO(3)-pairs with finite Fourier series on Riemannian surfaces, providing a full classification of transparent pairs in negatively curved cases using Bäcklund transformations.

Contribution

It introduces a classification of transparent pairs on Riemannian surfaces, especially in negatively curved cases, via Bäcklund transformations, advancing understanding of such geometric structures.

Findings

Complete classification of SO(3)-transparent pairs on negatively curved surfaces.

Use of Bäcklund transformations to characterize cohomologically trivial pairs.

Identification of conditions for finite Fourier series in the classification.

Abstract

Let $M$ be a closed orientable Riemannian surface. Consider an SO(3)-connection $A$ and a Higgs field $\Phi:M\to so(3)$. The pair $(A,\Phi)$ naturally induces a cocycle over the geodesic flow of $M$. We classify (up to gauge transformations) cohomologically trivial pairs $(A,\Phi)$ with finite Fourier series in terms of a suitable B\"acklund transformation. In particular, if $M$ is negatively curved we obtain a full classification of SO(3)-transparent pairs.