DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

86 Chapter 3. Surfaces: Further Topicsa. Calculate the geodesic curvature of the circle u = u 0 and compute∫κ g ds.∂M rb. Calculate χ(M r ).∫∫c. Use the Gauss-Bonnet Theorem to compute KdA. Find the limit as r →∞. (ThisM ris the total curvature of the pseudosphere.)∫∫d. Calculate the area of M r directly, and use this to deduce the value of KdA.e. Explain the relation between the total curvature and the image of the Gauss map of M.11. Check the details of the proof of Theorem 1.5 in the presence of corners. (See Exercise 1.3.6.)12. a. Use Corollary 1.3 to prove that M is flat if and only if the holonomy around all (“small”)closed curves that bound a region in M is zero.b. Show that even on a flat surface, holonomy can be nontrivial around certain curves.13. Reprove the result of Exercise 2.3.13 by considering the holonomy around a (sufficiently small)quadrilateral formed by four of the lines. Does the result hold if there are two families ofgeodesics in M always intersecting at right angles?14. Suppose α is a closed space curve with κ ≠0. Assume that the normal indicatrix (i.e., thecurve traced out on the unit sphere by the principal normal) is a simple closed curve in theunit sphere. Prove then that it divides the unit sphere into two regions of equal area. (Hint:Apply the Gauss-Bonnet Theorem to one of those regions.)15. Suppose M ⊂ R 3 is a closed, oriented surface with no boundary with K>0. It follows thatM is topologically a sphere (why?). Prove that M is convex; i.e., for each P ∈ M, M lies ononly one side of the tangent plane T P M. (Hint: Use the Gauss-Bonnet Theorem and Gauss’soriginal interpretation of curvature indicated in the remark on p. 48 to show the Gauss mapmust be one-to-one. Then look at the end of the proof of Theorem 3.4 of Chapter 1.)M2. An Introduction to Hyperbolic GeometryHilbert proved in 1901 that there is no surface (without boundary) in R 3 with constant negativecurvature with the property that it is a closed subset of R 3 (i.e., every Cauchy sequence of pointsin the surface converges to a point of the surface). The pseudosphere fails the latter condition.Nevertheless, it is possible to give a definition of an “abstract surface” (not sitting inside R 3 )together with a first fundamental form. As we know, this will be all we need to calculate Christoffelsymbols, curvature (Theorem 3.1 of Chapter 2), geodesics, and so on.Definition. The hyperbolic plane H is defined to be the half-plane {(u, v) ∈ R 2 : v > 0},equipped with the first fundamental form I given by E = G =1/v 2 , F =0.Now, using the formulas (‡) onp.55, we find thatΓ u uu = E u2E =0Γv uu = − E v2G = 1 v