Study your class notes and read Sections 6.5 and 6.6 and solve the following problems.
Section 6.5: 3 (remember that in the text, the curve is assumed to be smooth at all points, including 0 = 2 pi)
Section 6.6: 2 (consists of a few small problems), 4, 5, 6
1. Show that if there exist two simple closed geodesics on a compact connected surface of positive curvature (i.e., positive Gaussian curvature), then they intersect.
2. Show that the sum of the interior angles of a geodesic triangle is equal to pi, if K=0; greater than pi, if K > 0; and less than pi, if K < 0.
3. Compute the Euler characteristic of the surface x^2 + y^4 + z^6 = 1. [Hint: construct a homeomorphism between it and the unit sphere.]