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Unformatted text preview: Let p be a point of a surface M , T be a geodesic triangle which contains p , and α , β , γ be the angles of T . Show that K ( p ) = lim T → p α + β + γ-π Area ( T ) . In particular, note that the above proves Gauss’s Theorema Egregium. 1 Last revised: December 8, 2004 1 Exercise 8. Show that the sum of the angles of a geodesic triangle on a surface of positive curvature is more than π , and on a surface of negative curvature is less than π . Exercise 9. Show that on a simply connected surface of negative curvature two geodesics emanating from the same point will never meet. Exercise 10. Let M be a surface homeomorphic to a sphere in R 3 , and let Γ ⊂ M be a closed geodesic. Show that each of the two regions bounded by Γ have equal areas under the Gauss map. Exercise 11. Compute the area of the pseudo-sphere, i.e. the surface of revolution obtained by rotating a tractrix. 2...
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