Reading: Chapter 8: pp. 165-169; Chapter 9: pp. 171-173.
Problems: Chapter 8: 5, 7, 9, 11.
Chapter 9: 1 (It should be "locally isometric" in the problem, rather than "isometric". Otherwise, the plane would have been a surface of revolution. Use Theorem 9.4 and the computation of the first fundamental form (and thereby the line element) in the example after Corollary 8.9. If you use the arc length parametrization of the generating curve in that example, the formula will be simpler. The rest is to see if you can change your coordinates on the given surface so as to make the line element change from Λ(u) (du2 + dv2) to something like du2 + Λ(u) dv2. See p. 132 on how the components of the metric change with changes of coordinates. Adjust your coordinates, if necessary, to make sure you are able to find λ and μ as in the first fundamental form of a general surface of revolution. Working with unit-speed curves is always simpler, and so is looking for a surface of revolution generated by a unit-speed curve. Skip the conformal mapping question.)
Last modified: (2013-03-13 16:32:54 CDT)