[Math] Chern-Gauss-Bonnet theorem for orbifolds

Question

There's a Gauss-Bonnet theorem for compact 2-orbifolds(due to Satake, I think), which gives a relation between the curvature of a Riemannian orbifold and the orbifold topology(i.e. taking into account not just the structure of O as a Hausdorff topological space, but also the structure of the singular points.) The only proof that I've seen was very much like the classical one of the Gauss-Bonnet theorem, with geodesic triangulations, angle defects and so on. This approach doesn't generalise to higher dimensions to prove the Chern-Gauss-Bonnet theorem, and I haven't even found a conjectured Chern-Gauss-Bonnet formula. I'm especially interested in the four-dimensional case, where the integrand is amenable. One reason I'm interested is just to get a feel for how much more complicated orbifolds are than manifolds; on the one hand, many basic definitions seem to go through from the manifold world to the orbifold world, but the generalisation leads to significant complications in practice. On the other hand, Kleiner and Lott posted a paper on the arxiv in which they use Ricci flow to geometrise 3-orbifolds, so orbifolds certainly seem like a good arena in which to generalise differential geometry. Chern-Gauss-Bonnet is a bit of a benchmark for higher-dimensional Riemannian geometry, and I'd like to know the state of the art is in this case.

Answer 1

Yes, there is a Gauss-Bonnet-Chern theorem for orbifolds, under a certain technical restriction. The proof is by reduction to the classical case, extending all definitions in the only meaningful way. It has been proven by Satake a long time ago:

http://projecteuclid.org/DPubS/Repository/1.0/Disseminate?view=body&id=pdf_1&handle=euclid.jmsj/1261153826

Here is an outline, in my own terms. For me, an orbifold $O$ is a smooth Deligne-Mumford stack (see here: http://arxiv.org/pdf/math/0203100). There is the coarse moduli space $|O|$. If $O$ is a quotient $M//G$ of some proper action of a discrete group, then $|O|$ is the space of orbits. In the definition of orbifolds that is used in $3$-dimensional topology, $|O|$ would be the underlying space which has some atlasses. On orbifolds, we have the following structures available: there is the tangent bundle $TO \to O$; $TO$ is an orbifold as well; this is a vector bundle in the category of stacks; the quotient $|TO| \to |O|$ is not quite a vector bundle.

Let me make the following assumption: There exists a compact oriented manifold $N$ and a finite covering $N \to O$ of degree $d$.

Under these circumstances, we can define the Euler number of the orbifold $\chi(O) \in \mathbb{Q}$ as follows:

$$\chi(O) := \frac{1}{d} \chi(N).$$

Using the multiplicativity of the ordinary Euler number, it is easy to see that $\chi(O)$ only depends on $O$ and not on $N$. Now I define the geometric side of the G-B-C theorem. Orbifolds have tangent bundles and we can talk about differential forms on orbifolds, de Rham complex, connnections on vector bundles and the Levi-Civitta connection in the same way as for manifolds. The cleanest way to define connections might be the framework of connections on principal bundles. The last piece of differential geometry needed is Chern-Weil theory. Conclusion: to any Riemann metric on an $2n$-dimensional oriented orbifold (i.e., the tangent bundle is oriented), we get an Euler-Form $e$, a closed $2n$-form.

This form represents an element in the cohomology of the orbifold, but we have to say what this means. If $O=M //G$ were a global quotient, then $H^* (O) := H^* (EG \times_G M)$, the Borel-equivariant cohomology. If $O$ is not a global quotient, you have to replace the Borel space by the homotopy type of the orbifold.

How do you integrate the $2n$-form? Well, given a closed oriented $2n$-manifold $N$ and a map $f:N \to O$ as before, we define

$$\int_O e := \frac{1}{d} \int_N f^* e,$$

which is the only reasonable way to define integration over orbifolds. Since $N$ is a manifold, we can apply the classical Gauss-Bonnet-Chern theorem and find that

$$\chi(O)=\int_O e.$$