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LectureNotes16U - Let p be a point of a surface M T be a...

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Math 497C Dec 8, 2004 1 Curves and Surfaces Fall 2004, PSU Lecture Notes 16 2.14 Applications of the Gauss-Bonnet theorem We talked about the Gauss-Bonnet theorem in class, and you may find the statement and prove of it in Gray or do Carmo as well. The following are all simple consequences of the Gauss-Bonnet theorem: Exercise 1. Show that the sum of the angles in a triangle is π . Exercise 2. Show that the total geodesic curvature of a simple closed planar curve is 2 π . Exercise 3. Show that the Gaussian curvature of a surface which is home- omorphic to the torus must alwasy be equal to zero at some point. Exercise 4. Show that a simple closed curve with total geodesic curvature zero on a sphere bisects the area of the sphere. Exercise 5. Show that there exists at most one closed geodesic on a cylinder with negative curvature. Exercise 6. Show that the area of a geodesic polygon with k vertices on a sphere of radius 1 is equal to the sum of its angles minus ( k - 2) π . Exercise 7.
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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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