On Kanai's conjecture for frame flows over negatively curved manifolds

2025/09/11 by Beaufort, Louis-Brahim
#37D30 #37D40 #57R22 #Differential Geometry (math.DG) #Dynamical Systems (math.DS) #FOS: Mathematics

paper · doi:10.48550/arxiv.2509.09500

Abstract

Let M be a closed, negatively curved Riemannian manifold of dimension n ≠ 4, 8 with strictly 1/4-pinched sectional curvature. We prove, that if the frame flow is ergodic and the sum of its unstable and stable bundles together with its flow direction is C², then M is homothetic to a real hyperbolic manifold. This extends to higher dimensions a previous result of Kanai in dimension 3. The proof generalises to isometric extensions of geodesic flows to a principal bundle P with compact structure group and yields the following alternative : either P is flat, or M is hyperbolic.

Citations

• Semiclassical analysis on principal bundles
• Rigidity of flat holonomies
• Measurable rigidity of the cohomological equation for linear cocycles\n over hyperbolic systems
• Rigidity of equality of Lyapunov exponents for geodesic flows
• Cocycles with one exponent over partially hyperbolic systems
• Almost all cocycles over any hyperbolic system have nonvanishing Lyapunov exponents

Related

• Isospectral connections, ergodicity of frame flows, and polynomial maps between spheres
• Consequences of ergodic frame flow for rank rigidity in negative curvature
• 2-Frame flow dynamics and hyperbolic rank rigidity in nonpositive curvature
• Closed, negatively curved Riemannian manifolds with a flat horosphere are hyperbolic
• Ergodicity of the geodesic flow for groups with a contracting element