DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

58 Chapter 2. Surfaces: Local TheoryFor example, the plane and cylinder are locally isometric, and hence the cylinder (as we wellknow) is flat. We now conclude that since the Gaussian curvature of a sphere is nonzero, a spherecannot be locally isometric to a plane. Thus, there is no way to map the earth “faithfully” (preservingdistance)—even locally—on a piece of paper. In some sense, the Mercator projection (seeExercise 2.1.9) is the best we can do, for, although it distorts distances, it does preserve angles.Let’s now apply the Codazzi equations to prove a rather striking result about surfaces withK =0everywhere, called flat surfaces.Proposition 3.3. Suppose M is a flat surface with no planar points. Then M is a ruled surfacewhose tangent plane is constant along the rulings.Proof. Since M has no planar points, we can choose k 1 =0and k 2 ≠0everywhere. Thenby Theorem 3.3 of the Appendix, there is a local parametrization of M so that the u-curves arethe first lines of curvature and the v-curves are the second lines of curvature. This means first ofall that F = m =0. Now, since k 1 =0,for any P ∈ M we have S P (x u )=0, and so n u = 0everywhere and n is constant along the u-curves.We now want to show that the u-curves are in fact lines. First, l =II(x u , x u )=−S P (x u )·x u =0and n ≠0,since k 2 ≠0.From the first of the Codazzi equations we now deduce thatand so Γ v uu =0. This means that0=l v − m u = lΓ u uv + m ( Γ v uv − Γ u uu)− nΓvuu = −nΓ v uu,x uu =Γ u uux u +Γ v uux v + ln =Γ u uux uis just a multiple of x u .Thus, the tangent vector x u never changes direction as we move along theu-curves, and this means that the u-curves must be lines. In conclusion, we have a ruled surfacewhose tangent plane is constant along rulings. □Remark. Such ruled surfaces are called developable. (See Exercise 10 and Exercise 2.1.8.) Theterminology comes from the fact that they can be rolled out—or “developed”—onto a plane.Next we prove a striking global result about compact surfaces. (Recall that a subset of R 3 iscompact if it is closed and bounded. The salient feature of compact sets is the maximum valuetheorem: A continuous real-valued function on a compact set achieves its maximum and minimumvalues.) We begin with a straightforwardProposition 3.4. Suppose M ⊂ R 3 is a compact surface. Then there is a point P ∈ M withK(P ) > 0.Proof. Because M is compact, the continuous function f(x) =‖x‖ achieves its maximum atsome point of M, and so there is a point P ∈ M farthest from the origin (which may or may not beinside M), as indicated in Figure 3.2. Let f(P )=R. AsExercise 1.2.7 shows, the curvature of anycurve α ⊂ M at P is at least 1/R, so—if we choose the unit normal n to be inward-pointing—everynormal curvature of M at P is at least 1/R. Itfollows that K(P ) ≥ 1/R 2 > 0. (That is, M is atleast as curved at P as the circumscribed sphere of radius R tangent to M at P .) □