The Gauss-Bonnet theorem for Cayley-Klein geometries of dimension two (Q851543)

The goal of the paper is to state and proof a Gauss-Bonnet theorem for a polygonal region \(M\) in one of the nine two-dimensional Cayley-Klein geometries. The notion of a polygonal region and the geodesic curvature \(k_{g}\) of its boundary \(\Gamma\) is readily available. The author defines the discontinuity \(\theta_{j}\) at the vertex \(P_{j}\) of the boundary, including the case of the space-time. The main result (the Gauss-Bonnet formula) states that \(\int_{\Gamma}k_{g}\,ds+\sum_{j}\theta_{j}+ \iint_{M} K\,dA=2\pi\), for the non-space-times, and \(\int_{\Gamma}k_{g}\,ds+\sum_{j} \theta_{j}+ \iint_{M} K\,dA=0\), for space-times.