IMPORTANT PROBLEMS FROM THE TEXT

Remark: most of these were assigned as homework. Those that weren't
are marked (E).

p.151

5. mean curvature= ave. of normal curvatures over the unit circle of directions
2. If a surface is tangent to a plane along a curve, the points of this curve are
parabolic or planar points of the surface (E).
8. Image of the Gauss map of certain surfaces.
15.Theorem of Joachimstahl (E)
17. The Gauss map of a minimal surface is conformal

p.168

6. the pseudosphere
7. surfaces of revolution with constant curvature
13. change in K and H under a homothety of R^3.
16. A compact surface must have an ellipic point (E) (see also p. 321)
20. umbilic points of a generic ellipsoid.

p.230

11. distance-preserving maps of R^3 which map S to S restrict to
isometries of S. Ex: orthogonal transformations and the sphere.
12. Isometry of the cylinder fixing exactly two points (E)
14. A map is locally conformal if and only if it preserves angles (E)
15. A pair of differentiable functions satisfying the Cauchy-Riemann
equations defines a conformal map away from the critical set. (E)
18. Area-preserving plus conformal implies isometry.
19. An area-preserving diffeo. from the sphere (minus 2 points) to the cylinder.

p.237

2. Gauss curvature in isothermal coordinates.
3. A diffeo. which preserves Gauss curvature is not necessarily an isometry.
5. Gauss curvature for a Tchebyshef net.
6,7. Non-existence of certain pairs (I,II).
9. examples of surfaces not pairwise locally isometric (Gauss's theorem).

p.212

12. Isothermal parameters on a minimal surface without umbilics
13. There are no compact minimal surfaces
14. Conjugate minimal surfaces

p. 260(E)

5,6 parallels of a torus of revolution
15.parallel transport on the sphere

p.282(E)

1. K takes all signs on a compact orientable surface
4.Euler characteristic computations
5.parallel transport and curvature
6. index of vector fields in the plane
9. vector fields on disks (transverse to boundary) have singular points.