The frame flow of negatively curved manifolds is one of the first examples of partially hyperbolic dynamics: it is defined as an extension of the geodesic flow (which is uniformly hyperbolic) to the frame bundle of the manifold by means of parallel transport. The study of its statistical properties such as ergodicity, mixing or exponential mixing, is still widely open. We have recently shown that, under reasonable pinching assumptions on the sectional curvature, the frame flow of even-dimensional manifolds is ergodic and mixing — the odd-dimensional case being already solved by Brin-Gromov in 1980.
Surprisingly, this problem is linked to questions of a very diverse nature: — in algebraic topology: the classification of the reductions of the structure groups of the sphere, — in Riemannian geometry: the existence of almost-parallel structures (e.g. Killing forms, almost-Kähler or almost-G_2 structures, etc.) — in algebraic geometry: the classification of polynomial maps between spheres. The purpose of this research project is to explore these multiple aspects of the problem and to progress on our understanding of the frame flow.