The Holonomy Inverse Problem

TL;DR

This paper proves that the primitive trace map, which encodes holonomy traces along primitive closed geodesics, is locally injective for generic connections on Anosov manifolds of dimension at least 3, with applications to spectral rigidity.

Contribution

It establishes local injectivity of the primitive trace map near generic points for higher-dimensional Anosov manifolds and explores global cases under specific conditions, introducing new dynamical and geometric tools.

Findings

Primitive trace map is locally injective near generic points in dimension ≥ 3.

Global injectivity results for flat bundles, line bundles, and certain negative curvature cases.

Spectral rigidity for the connection Laplacian derived from local injectivity.

Abstract

Let $(M,g)$ be a smooth Anosov Riemannian manifold and $\mathcal{C}^\sharp$ the set of its primitive closed geodesics. Given a Hermitian vector bundle $\mathcal{E}$ equipped with a unitary connection $\nabla^{\mathcal{E}}$, we define $\mathcal{T}^\sharp(\mathcal{E}, \nabla^{\mathcal{E}})$ as the sequence of traces of holonomies of $\nabla^{\mathcal{E}}$ along elements of $\mathcal{C}^\sharp$. This descends to a homomorphism on the additive moduli space $\mathbb{A}$ of connections up to gauge $\mathcal{T}^\sharp: (\mathbb{A}, \oplus) \to \ell^\infty(\mathcal{C}^\sharp)$, which we call the $\textit{primitive trace map}$. It is the restriction of the well-known $\textit{Wilson loop}$ operator to primitive closed geodesics. The main theorem of this paper shows that the primitive trace map $\mathcal{T}^\sharp$ is locally injective near generic points of $\mathbb{A}$ when $\dim(M) \geq 3$. We obtain global results in some particular cases: flat bundles, direct sums of line bundles, and general bundles in negative curvature under a spectral assumption which is satisfied in particular for connections with small curvature. As a consequence of the main theorem, we also derive a spectral rigidity result for the connection Laplacian. The proofs are based on two new ingredients: a Liv\v{s}ic-type theorem in hyperbolic dynamical systems showing that the cohomology class of a unitary cocycle is determined by its trace along closed primitive orbits, and a theorem relating the local geometry of $\mathbb{A}$ with the Pollicott-Ruelle resonance near zero of a certain natural transport operator.