On Frame Flow Ergodicity

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.