On uniqueness of differential structures on orbifolds

Abstract

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