## Kodai Mathematical Journal

- Kodai Math. J.
- Volume 37, Number 2 (2014), 427-433.

### New proofs of theorems of Kathryn Mann

#### Abstract

We give a shorter proof of the following theorem of Kathryn Mann [**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 *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 *C*^{r} diffeomorphisms of the circle to the group of *C*^{p} diffeomorphisms of the circle is the conjugation of the standard inclusion by a *C*^{p} diffeomorphism, if *r* ≥ *p*, *r* ≠ 2 and *p* ≠ 1.

#### Article information

**Source**

Kodai Math. J., Volume 37, Number 2 (2014), 427-433.

**Dates**

First available in Project Euclid: 3 July 2014

**Permanent link to this document**

https://projecteuclid.org/euclid.kmj/1404393896

**Digital Object Identifier**

doi:10.2996/kmj/1404393896

**Mathematical Reviews number (MathSciNet)**

MR3229085

**Zentralblatt MATH identifier**

1297.57069

#### Citation

Matsumoto, Shigenori. New proofs of theorems of Kathryn Mann. Kodai Math. J. 37 (2014), no. 2, 427--433. doi:10.2996/kmj/1404393896. https://projecteuclid.org/euclid.kmj/1404393896