Reading: Your lecture notes.
Problem P: Show that for a surface given as the graph of a function x = f (y,z), the Gaussian curvature in the (y,z) parametrization is given by K = (fyyfzz - fyz2)/(1 + fy2 + fz2)2. [Hint: Do this by computing the first partials xy and xz and the second partials, the normal vector xy×xz and its normalization N, and then the first and the second fundamental forms and after that the quotient of their determinants, like we did it in class on Wednesday, April 17, for the saddle surface z = x2 - y2. See that computation of N on p. 161. Note that the text suggests a different computation of dNp and thereby K = det dNp there, which I believe to be more complicated. What we did in class rather goes via Corollary 8.9 on p. 166.]
Problem Q: Show that the sheet { (x,y,z) | - x2 + y2 + z2 = - R2, x > 0 } of the two-sheeted hyperboloid as a surface in the Euclidean space R3 has positive Gaussian curvature. [Hint: Use the previous problem.]
Problem R: Show that the first fundamental form induced on the pseudosphere +S12 = { (x,y,z) | - x2 + y2 + z2 = - R2, x > 0} from the pseudo-Euclidean metric ⟨(x,y,z), (x,y,z)⟩ = - x2 + y2 + z2 is positive definite, i.e., for any nonzero tangent vector v to +S12, we have ⟨v, v⟩ > 0. [Hint: One way to do it might be by computing the metric coefficients E, F, and G and checking that E > 0 and EG - F2 > 0.]
Problem S: Prove that the stereographic projection of the sphere S2 to the plane takes circles to circles or lines.
Problem T: Show that in the Poincaré model of hyperbolic geometry, every two distinct points are incident to (i.e., lie on) a unique line.
Last modified: (2013-04-20 02:02:51 CDT)