Anosov vector fields and Fried sections

TL;DR

This paper constructs a canonical nonzero section of the determinant line bundle associated with Anosov vector fields on compact manifolds and explores its smoothness and flatness properties in families and fibrations.

Contribution

It introduces a canonical section of the determinant line bundle linked to Anosov vector fields and analyzes its regularity and flatness in various geometric contexts.

Findings

The section is $C^{1}$ in families.

The section is flat when the bundle is flat on the total space.

The construction applies to flat vector bundles over compact manifolds.

Abstract

The purpose of this paper is to prove that if $Y$ is a compact manifold, if $Z$ is an Anosov vector field on $Y$, and if $F$ is a flat vector bundle, there is a corresponding canonical nonzero section $\tau_{\nu}\left(i_{Z}\right)$ of the determinant line $\nu=\det H\left(Y,F\right)$. In families, this section is $C^{1}$ with respect to the canonical smooth structure on $\nu$. When $F$ is flat on the total space of the corresponding fibration, our section is flat with respect to the Gauss-Manin connection on $\nu$.