On Lie algebras of vector fields on smooth orbifolds (Q1822105)

A paracompact Hausdorff space B is called an orbifold if an open covering \(\{U_ i\}\) of B closed under intersections exists such that for each \(U_ i\), there is an action of a finite group \(\Gamma_ i\) on an open subset \(V_ i\) of \({\mathbb{R}}^ n\) and homeomorphisms \(\Phi_ i: U_ i\to V_ i/\Gamma_ i\) satisfying the obvious compatibility conditions on the intersections \(U_ i\cap U_ j\). The author introduces the Lie algebra \({\mathcal D}(B)\) (\({\mathcal X}(B))\) of smooth vector fields on orbifolds and proves that the Lie algebra isomorphisms \({\mathcal D}(B)\to {\mathcal D}(B')\) (\({\mathcal X}(B)\to {\mathcal X}(B'))\) are in one-to-one correspondence with the diffeomorphisms \(B\to B'\) between orbifolds. The fibered orbifolds and the fiber preserving vector fields are investigated also.