DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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80 Chapter 3. Surfaces: Further TopicsK=0K=1 K=−1Figure 1.4is negatively curved, it takes less. Intuitively, this is because geodesics seem to “bow out” whenK>0 and “bow in” when K
§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
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DIFFERENTIALGEOMETRY:A First Course
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4 Chapter 1. Curves2πb2πaFigure 1
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6 Chapter 1. CurvesAlternatively, s
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8 Chapter 1. CurvesAn important obs
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10 Chapter 1. Curvesof the road, st
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12 Chapter 1. CurvesSummarizing,κ(
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14 Chapter 1. Curvescurvature κ. I
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16 Chapter 1. CurvesFigure 2.2Conve
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18 Chapter 1. Curvesalso Exercise 2
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20 Chapter 1. Curvesto the curve β
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22 Chapter 1. Curvesthe osculating
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24 Chapter 1. CurvesWe turn now to
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26 Chapter 1. Curvesintersect C eit
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28 Chapter 1. Curvesh(s,t)α(t)α(s
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Page 120: 118 INDEXisometry, 107K, 48k-point
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DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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