On uniqueness of differential structures on orbifolds

Abstract

It is known that every ${\mathrm{C}}^r$-orbifold, $1\le r\le \infty$, has a compatible ${\mathrm{C}}^s$-differential structure, for every $s$, where $r<s\le \omega$. We prove that if two reduced ${\mathrm{C}}^r$-orbifolds, $2\le r\le \omega$, are ${\mathrm{C}}^2$-diffeomorphic, then they are ${\mathrm{C}}^r$-diffeomorphic. It follows that the compatible ${\mathrm{C}}^s$-differential structure on a reduced ${\mathrm{C}}^r$-orbifold, $2\le r<s\le \omega$, is unique up to a ${\mathrm{C}}^s$-diffeomorphism.