Anosov Vector Fields and Fried Sections

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 \(\nu \).