Partially hyperbolic flows on flat vector bundles with an application to complete affine manifolds

Let $N$ be a manifold of dimension $m$ with a flat vector bundle given by a representation $\rho :{\pi }_1(N)\to \mathrm{G}\mathrm{L}(n,\mathbf{R})$ where ${\pi }_1(N)$ is finitely generated. The holonomy group $\rho$ is a $k$-partially hyperbolic holonomy representation if the flat bundle pulled back over the unit tangent bundle of a sufficiently large compact submanifold of $N$ splits into expanding, neutral, and contracting subbundles along the geodesic flow, where the expanding and contracting subbundles are $k$-dimensional with $k<n\mathrm{/}2$. Suppose that each element of $\rho ({\pi }_1(N))$ has an eigenvalue of norm $1$, or, alternatively, $\rho$ has some singular values of subexponential growth in terms of word length. We show that $\rho$ is a $P$-Anosov representation for a parabolic subgroup $P$ of $\mathrm{G}\mathrm{L}(n,\mathbf{R})$ if and only if $\rho$ is a partially hyperbolic representation. We are going to primarily employ representation theory techniques. As an application, we will show that the equivalence holds when $N$ is a complete affine $n$-manifold, and $\rho$ is a linear part of the holonomy representation. This had never been done over the full general linear group.