DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§1. Holonomy and the Gauss-Bonnet Theorem 81nFigure 1.6(i) ∆ λ is the image of a triangle under an (orientation-preserving) orthogonal parametrization;(ii) ∆ λ ∩ ∆ µ is either empty, a single vertex, or a single edge;(iii) when ∆ λ ∩ ∆ µ consists of a single edge, the orientations of the edge are opposite in ∆ λand ∆ µ ; and(iv) at most one edge of ∆ λ is contained in the boundary of M.We now make a standardDefinition. Given a triangulation T of a surface M with V vertices, E edges, and F faces, wedefine the Euler characteristic χ(M,T) =V − E + F .Example 4. We can triangulate a disk as shown in Figure 1.7, obtaining χ =1. Without∆ 2∆ 1∆ 3 ∆ 4V−E+F = 9−18+10 = 1V−E+F = 5−8+4 = 1Figure 1.7being so pedantic as to require that each ∆ λ be the image of a triangle under an orthogonalparametrization, we might just think of the disk as a single triangle with its edges puffed out; thenwe would have χ = V − E + F =3− 3+1=1,aswell. We leave it to the reader to triangulate asphere and check that χ(Σ, T) =2. ▽The beautiful result to which we’ve been headed is now the followingTheorem 1.7 (Global Gauss-Bonnet). Let M be an oriented surface with piecewise-smoothboundary, equipped with a triangulation T as above. If ɛ k , k =1,...,l, are the exterior angles of∂M, then∫∂M∫∫κ g ds + KdA+Ml∑ɛ k =2πχ(M,T).Proof. As we illustrate in Figure 1.8, we will distinguish vertices on the boundary and in theinterior, denoting the respective total numbers by V b and V i . Similarly, we distinguish among edgesk=1