  We show that on closed negatively curved Riemannian manifolds with simple
length spectrum, the spectrum of the Bochner Laplacian determines both the
isomorphism class of the vector bundle and the connection up to gauge under a
low-rank assumption. We also show that flows of frames on low-rank frame
bundles extending the geodesic flow in negative curvature are ergodic whenever
the bundle admits no holonomy reduction. This is achieved by exhibiting a link
between these problems and the classification of polynomial maps between
spheres in real algebraic geometry.
