DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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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 .) □
§3. The Codazzi and Gauss Equations and the Fundamental Theorem of Surface Theory 590MPFigure 3.2The reader is asked in Exercise 16 to find surfaces of revolution of constant curvature. Thereare, interestingly, many nonobvious examples. However, if we restrict ourselves to smooth surfaces,we have the following beautifulTheorem 3.5 (Liebmann). If M is a smooth, compact surface of constant Gaussian curvatureK, then K>0 and M must be a sphere of radius 1/ √ K.We will need the followingLemma 3.6 (Hilbert). Suppose P is not an umbilic point and k 1 (P ) >k 2 (P ). Suppose k 1 hasalocal maximum at P and k 2 has a local minimum at P . Then K(P ) ≤ 0.Proof. We work in a “principal” coordinate parametrization, 4 so that the u-curves are linesof curvature with principal curvature k 1 and the v-curves are lines of curvature with principalcurvature k 2 . Then we have k 1 = l/E, k 2 = n/G, and F = m =0.Bythe Codazzi equations andthe equations (‡) onp.55,we have(⋆)l v = 1 2 E v(k 1 + k 2 ) and n u = 1 2 G u(k 1 + k 2 ).Since l = k 1 E,wehavel v =(k 1 ) v E + k 1 E v ; similarly, n u =(k 2 ) u G + k 2 G u . Using the equations(⋆), we then obtain(k 1 ) v = E v2E (k 2 − k 1 ) and (k 2 ) u = G u(⋆⋆)2G (k 1 − k 2 );since k 1 ≠ k 2 and (k 1 ) v =(k 2 ) u =0atP ,weinfer that E v = G u =0atP .Differentiating the equations (⋆⋆), and remembering that (k 1 ) u =(k 2 ) v =0atP as well, wehave at P :(k 1 ) vv = E vv2E (k 2 − k 1 ) ≤ 0 (because k 1 has a local maximum at P )(k 2 ) uu = G uu2G (k 1 − k 2 ) ≥ 0 (because k 2 has a local minimum at P ),4 Since locally there are no umbilic points, the existence of such a parametrization is an immediate consequenceof Theorem 3.3 of the Appendix.
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DIFFERENTIALGEOMETRY:A First Course
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4 Chapter 1. Curves2πb2πaFigure 1
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6 Chapter 1. CurvesAlternatively, s
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108 Appendix. Review of Linear Alge
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ANSWERS TO SELECTED EXERCISES1.1.1
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116 SELECTED ANSWERS2.4.8 Only circ
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118 INDEXisometry, 107K, 48k-point
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DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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