A remark on local rigidity of conformal actions on the sphere
The first result of this paper is: Let \(M^{n+1}\) be an orientable, closed, hyperbolic manifold of dimension at least 3. Let \(\rho_0\) denote the standard action of the fundamental group of \(M\) on the \(n\)-sphere. A \(C^\infty\) action \(\rho\) of the fundamental group of \(M\) which is sufficiently \(C^2\)-close to \(\rho_0\) is \(C^\infty\) conjugate to \(\rho_0\). The second result states: Let \(\varphi^t_0\) denote the geodesic flow of \(M\) which is defined on the unit tangent bundle \(V\) and is an Anosov flow. Let \(f^+_0\) denote its weak unstable foliation. Then for any \(C^\infty\) foliation \(f^+\) of \(V\) of dimension \(n+1\), there exists a diffeomorphism of \(V\) onto itself that maps \(f^+\) to \(f_0^+\).
scientific article; zbMATH DE number 1489523