DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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42 Chapter 2. Surfaces: Local Theoryφ = f(u) and θ = v. Show that conformality and symmetry about the equator will dictatef(u) =2arctan(e −u ). Deduce that(Cf. Example 2 in Section 1 of Chapter 1.)x(u, v) =(sechu cos v, sechu sin v, tanh u).10. A parametrization x(u, v) iscalled a Tschebyschev net if the opposite sides of any quadrilateralformed by the coordinate curves have equal length.a. Prove that this occurs if and only if ∂E∂v = ∂G =0. (Hint: Express the length of the∂uu-curves, u 0 ≤ u ≤ u 1 ,asanintegral and use the fact that this length is independent ofv.)b. Prove that we can locally reparametrize by ˜x(ũ, ṽ) soastoobtain Ẽ = ˜G =1, ˜F =cos θ(ũ, ṽ) (so that the ũ- and ṽ-curves are parametrized by arclength and meet at angle/θ). (Hint: Choose ũ as a function of u so that ˜xũ = x u (dũ/du) has unit length.)11. Suppose x and y are two parametrizations of a surface M near P .Sayx(u 0 ,v 0 )=P = y(s 0 ,t 0 ).Prove that Span(x u , x v )=Span(y s , y t ) (where the partial derivatives are all evaluated at theobvious points). (Hint: f = x −1 ◦y gives a C 1 map from an open set around (s 0 ,t 0 )toanopenset around (u 0 ,v 0 ). Apply the chain rule to show y s , y t ∈ Span(x u , x v ).)12. (A programmable calculator or Maple may be useful for parts of this problem.) A catenoid, aspictured in Figure 1.9, is parametrized byx(u, v) =(a cosh u cos v, a cosh u sin v, au), u ∈ R, 0 ≤ v0 fixed).Figure 1.9*a. Compute the surface area of that portion of the catenoid given by |u| ≤ 1/a. (Hint:cosh 2 u = 1 2(1 + cosh 2u).)b. Given the pair of parallel circles x 2 + y 2 = R 2 , |z| =1,for what values of R is there atleast one catenoid with the circles as boundary? (Hint: Graph f(t) =t cosh(1/t).)c. For the values of R in part b, compare the area of the catenoid(s) with 2πR 2 , the area ofthe pair of disks filling in the circles. For what values of R does the pair of disks have theleast area?13. There are two obvious families of circles on a torus. Find a third family. (Hint: Look for a planethat is tangent to the torus at two points. Using the parametrization of the torus, you should
§2. The Gauss Map and the Second Fundamental Form 43be able to find parametric equations for the curve in which the bitangent plane intersects thetorus.)2. The Gauss Map and the Second Fundamental FormGiven a regular parametrized surface M, the function n: M → Σ that assigns to each pointP ∈ M the unit normal n(P ), as pictured in Figure 2.1, is called the Gauss map of M. Asweshallsee in this chapter, most of the geometric information about our surface M is encapsulated in thePnn(P)Figure 2.1mapping n.Example 1. A few basic examples are these.(a) On a plane, the tangent plane never changes, so the Gauss map is a constant.(b) On a cylinder, the tangent plane is constant along the rulings, so the Gauss map sends theentire surface to an equator of the sphere.(c) On a sphere centered at the origin, the Gauss map is merely the (normalized) positionvector.(d) On a saddle surface (as pictured in Figure 2.1), the Gauss map appears to “reverse orientation”:As we move counterclockwise in a small circle around P ,wesee that the unit vectorn turns clockwise around n(P ). ▽Recall from the Appendix that for any function f on M (scalar- or vector-valued) and anytangent vector V ∈ T P M,wecan compute the directional derivative D V f(P )bychoosing a curveα: (−ε, ε) → M with α(0) = P and α ′ (0) = V and computing (f ◦α) ′ (0).To understand the shape of M at the point P , might try to understand the curvature at P ofvarious curves in M. Perhaps the most obvious thing to try is various normal slices of M. That is,we slice M with the plane through P spanned by n(P ) and a unit vector V ∈ T P M.Various suchnormal slices are shown for a saddle surface in Figure 2.2. Let α be the arclength-parametrizedcurve obtained by taking such a normal slice. We have α(0) = P and α ′ (0) = V. Then since thecurve lies the plane spanned by n(P ) and V, the principal normal of the curve at P must be ±n(P )(+ if the curve is curving towards n, − if it’s curving away). Applying Lemma 2.1 of Chapter 1
Page 1: DIFFERENTIALGEOMETRY:A First Course
Page 6 and 7: 4 Chapter 1. Curves2πb2πaFigure 1
Page 8 and 9: 6 Chapter 1. CurvesAlternatively, s
Page 10 and 11: 8 Chapter 1. CurvesAn important obs
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Page 14 and 15: 12 Chapter 1. CurvesSummarizing,κ(
Page 16 and 17: 14 Chapter 1. Curvescurvature κ. I
Page 18 and 19: 16 Chapter 1. CurvesFigure 2.2Conve
Page 20 and 21: 18 Chapter 1. Curvesalso Exercise 2
Page 22 and 23: 20 Chapter 1. Curvesto the curve β
Page 24 and 25: 22 Chapter 1. Curvesthe osculating
Page 26 and 27: 24 Chapter 1. CurvesWe turn now to
Page 28 and 29: 26 Chapter 1. Curvesintersect C eit
Page 30 and 31: 28 Chapter 1. Curvesh(s,t)α(t)α(s
Page 32 and 33: 30 Chapter 1. Curvesy( x(s) ,y(s))
Page 34 and 35: 32 Chapter 1. Curvesɛ2ɛ 1ɛ 3CFig
Page 36 and 37: CHAPTER 2Surfaces: Local Theory1. P
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92 Chapter 3. Surfaces: Further Top
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100 Chapter 3. Surfaces: Further To
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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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