Two Gauss-Bonnet and Poincar\'{e}-Hopf Theorems for Orbifolds with Boundary

Abstract

The goal of this work is to generalize the Gauss-Bonnet and Poincar\'{e}-Hopf Theorems to the case of orbifolds with boundary. We present two such generalizations, the first in the spirit of Satake. In this case, the local data (i.e. integral of the curvature in the case of the Gauss-Bonnet Theorem and the index of the vector field in the case of the Poincar\'{e}-Hopf Theorem) is related to Satake's orbifold Euler characteristic, a rational number which depends on the orbifold structure. For the second pair of generalizations, we use the Chen-Ruan orbifold cohomology to express the local data in a way which can be related to the Euler characteristic of the underlying space of the orbifold. This case applies only to orbifolds which admit almost-complex structures.