DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

§1. Holonomy and the Gauss-Bonnet Theorem 79e.g., by stereographic projection from the south pole (cf. Example 1(e) in Section 1 of Chapter 2).▽Remark. In more advanced courses, the holonomy around the closed curve α is interpretedas a rotation of the tangent plane of M at α(0). That is, what matters is ∆ψ (mod 2π), i.e.,the change in angle disregarding multiples of 2π. This quantity does not depend on the choice offraming e 1 , e 2 .We now set to work on one of the crowning results of surface theory.Theorem 1.5 (Local Gauss-Bonnet). Suppose R is a simply connected region with piecewisesmooth boundary in a parametrized surface. If ∂R has exterior angles ɛ j , j =1,...,s, then∫ ∫∫s∑κ g ds + KdA+ ɛ j =2π.∂RRj=1ɛ2ɛ 1ɛ 3CFigure 1.3Note, as we indicate in Figure 1.3, that we measure exterior angles so that |ɛ j |≤π for all j.Proof. If ∂R is smooth, then from our earlier discussion we infer that∫ ∫∫κ g ds + KdA =∆θ =2π.∂RRBut the unit tangent vector turns less by the amount ∑ sj=1 ɛ j,sothe result follows. (We ask thereader to check the details in Exercise 11.) □Corollary 1.6. Forageodesic triangle (i.e., a region whose boundary consists of three geodesicsegments) R with interior angles ι 1 , ι 2 , ι 3 ,wehave ∫∫ R KdA =(ι 1 + ι 2 + ι 3 ) − π, the angle excess.Proof. Since the boundary consists of geodesic segments, the geodesic curvature integral dropsout, and we are left with∫∫3∑3∑3∑KdA =2π − ɛ j =2π − (π − ι j )= ι j − π,Ras required.□j=1Remark. It is worthwhile to consider the three special cases K =0,K =1,K = −1, aspictured in Figure 1.4. When M is flat, the sum of the angles of a triangle is π, asinthe Euclideancase. When M is positively curved, it takes more than π for the triangle to close up, and when Mj=1j=1