We give a shorter proof of the following theorem of Kathryn Mann cite{M}: the identity component of the group of the compactly supported $C^r$ diffeomorphisms of $R^n$ cannot admit a nontrivial $C^p$-action on $S^1$, provided $ngeq2$, $r eq n+1$ and $pgeq2$. We also give a new proof of another theorem of Mann: any nontrivial endomorphism of the group of the orientation preserving $C^r$ diffeomorphisms of the circle is the conjugation by a $C^r$ diffeomorphism, if $rgeq3$.
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