The frame flow over negatively-curved Riemannian manifolds is a historical example of a partially hyperbolic dynamical system. Excluding some obvious counterexamples such as Kähler manifolds, its ergodicity was conjectured by Brin in the 70s. While it has been known since Brin-Gromov (1980) that it is ergodic on odd-dimensional manifolds (and dimension not equal to 7), the even-dimensional case is still open. In this talk, I will explain recent progress towards this conjecture: I will show that in dimensions 4k+2 the frame flow is ergodic if the Riemannian manifold is 0.27 pinched (i.e., the sectional curvature is between -1 and -0.27), and in dimensions 4k if it is 0.55 pinched. This problem turns out to be surprisingly rich and at the interplay of different fields: (partially) hyperbolic dynamical systems, algebraic topology (classification of topological structures over spheres), Riemannian geometry and harmonic analysis (Pestov identity and microlocal analysis). Joint work with Mihajlo Cekić, Andrei Moroianu, Uwe Semmelmann.
Fanny Kassel