New proofs of theorems of Kathryn Mann

We give a shorter proof of the following theorem of Kathryn Mann [M]: the identity component of the group of the compactly supported Cr diffeomorphisms of Rn cannot admit a nontrivial Cp-action on S1, provided n ≥ 2, r ≠ n + 1 and p ≥ 2. We also give a new proof of another theorem of Mann [M]: any nontrivial homomorphism from the group of the orientation preserving Cr diffeomorphisms of the circle to the group of Cp diffeomorphisms of the circle is the conjugation of the standard inclusion by a Cp diffeomorphism, if r ≥ p, r ≠ 2 and p ≠ 1.