2
$\begingroup$

It may a stupid question. As stated in Lang-Fang Wu' paper (JDG 33(1991)) “Bad orbifolds do not admit metrics of constant curvature. ” Can anyone give a proof or a reference about this statement. How about the case for high-dim orbifold? Thank you for any comments.

(An orbifold is called bad if it can not be covered by a manifold. For example, the only closed bad 2-orbifold are teardrops and (p,q)-footballs, for which they have only one singular point with order $n$ and two singular points with order $p,q$(which are coprime numbers) resp.)

$\endgroup$
3
  • $\begingroup$ It will help if you define a "bad orbifold". $\endgroup$ Mar 26, 2018 at 18:36
  • 3
    $\begingroup$ For 2-dimensional orbifolds I think this is Theorem 13.3.6 In Chapter 13 ( "Orbifolds") of "The Geometry and Topology of Three Manifolds" by W. P. Thurston. $\endgroup$
    – Nick L
    Mar 26, 2018 at 19:28
  • $\begingroup$ The idea is that if you have a metric of constant curvature, there exists the so-called developing "map" (a multi-valued map) whose inverse is a covering by a manifold (in 2-D case this manifold is one of the surfaces of constant curvature.) $\endgroup$ Mar 27, 2018 at 1:58

0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Browse other questions tagged or ask your own question.