In differential geometry, the Euler characteristic of an orbifold, or orbifold Euler characteristic, is a generalization of the topological Euler characteristic that includes contributions coming from nontrivial automorphisms. In particular, unlike a topological Euler characteristic, it is not restricted to integer values and is in general a rational number. It is of interest in mathematical physics, specifically in string theory. Given a compact manifold quotiented by a finite group , the Euler characteristic of is
where is the order of the group , the sum runs over all pairs of commuting elements of , and is the space of simultaneous fixed points of and . (The appearance of in the summation is the usual Euler characteristic.) If the action is free, the sum has only a single term, and so this expression reduces to the topological Euler characteristic of divided by .