Week 47
We have now reached one of our goals for this course: to characterize the classical geometries, at least locally, as Riemannian surfaces having constant curvature. The proof shows the power of geodesic polar coordinates, and these will again be important next week in the proof of our last, important theorem: the famous Gauss-Bonnet theorem.
After we finish the theory, I will go through some old exam problems. The last session will then be December 4.
NOTE: There is a particularly nasty misprint in the formula for the inner product in the middle of page 151 in the notes. The term (1-a2-b2)2 in the denominator should have been (1-x2-y2)2 , where the vector (a,b) lies in the tangent space of the Poincare disk at the point z=x+yi.
Problems for Tuesday November 25: 5.6.12, 5.7.1 and the following little challenge: Try to prove that every point p in a Riemannian surface S has a neigborhood V such that every point q in V can be joined to p by a unique geodesic which is shortest curve in S between p and q.