DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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116 SELECTED ANSWERS2.4.8 Only circles. By Exercise 2 such a curve will also have constant curvature, and byMeusnier’s formula, Proposition 2.5, the angle φ between N and n = x is constant.Differentiating x · N = cos φ = const yields τ(x · B) =0. Either τ =0,inwhich casethe curve is planar, or else x·B =0,inwhich case x = ±N, soτ = N ′ ·B = ±x ′ ·B =±T · B =0. (In the latter case, the curve is a great circle.)3.1.1 a. 2π sin u 03.1.3 a. 1, b.,c. 3.3.2.1 a. The semicircle centered at (2, 0) of radius √ 5; d(P, Q) =ln ( (3 + √ 5)/2 ) .3.2.4 Show first that for an arclength parametrization κ g = u ′ (s)/v(s)+θ ′ (s). Take a circleof the form α(t) = ( a + b cos t, b(sin t +1) ) and show υ(t) =‖α ′ (t)‖ =1/(sin t + 1).3.2.9 κ g = coth R3.3.4 We have κ n =II(e 1 , e 1 )=−de 3 (e 1 ) · e 1 = ω 13 (e 1 ). Since e 3 = sin θe 2 + cos θe 3 ,the calculations of Exercise 3 show that ω 13 = sin θω 12 + cos θω 13 , so ω 13 (e 1 ) =sin θω 12 (e 1 )=κ sin θ. Here θ is the angle between e 3 and e 3 ,sothis agrees with ourprevious result.3.3.7 We have ω 1 = bdu(and ω 2 = (a)+ b cos u) dv, so ω 12 = − sin udv and dω 12 =cos ucos u− cos udu∧ dv = −ω 1 ∧ ω 2 ,soK =b(a + b cos u)b(a + b cos u) .3.4.2 a. Taking ξ = f gives us ∫ 10 f(t)2 dt =0. Since f(t) 2 ≥ 0 for all t, iff(t 0 ) ≠0,wehavean interval [t 0 − δ, t 0 + δ] onwhich f(t) 2 ≥ f(t 0 ) 2 /2, and so ∫ 10 f(t)2 dt ≥ f(t 0 ) 2 δ>0.3.4.9 y = 1 2 cosh(2x)A.1.1A.1.2A.2.4Consider z = x − y. Then we know that z · v i =0,i =1, 2. Since {v 1 , v 2 } is a basisfor R 2 , there are scalars a and b so that z = av 1 + bv 2 . Then z · z = z · (av 1 + bv 2 )=a(z · v 1 )+b(z · v 2 )=0,soz = 0, asdesired.Hint: Take u =( √ a, √ b) and v =( √ b, √ a).Let v = ∫ ba f(t) dt. If v ≠ 0, we have ‖v‖2 = v · ∫ ba f(t) dt = ∫ bav · f(t) dt ≤∫ ba ‖v‖‖f(t)‖ dt = ‖v‖ ∫ ba‖f(t)‖ dt (using the Cauchy-Schwarz inequality u · v ≤‖u‖‖v‖), so ‖v‖ ≤ ∫ ba‖f(t)‖ dt, asneeded.
INDEXangle excess, 79arclength, 6asymptotic curve, 47, 49, 52asymptotic direction, 47, 51, 53Bertrand mates, 20binormal vector, 11Bäcklund, 100C k ,1,34, 109catenary, 5catenoid, 42, 63Cauchy-Schwarz inequality, 109chain rule, 109characteristic polynomial, 108Christoffel symbols, 54Codazzi equations, 57–60, 98compact, 58conformal, 39connection form, 99convex, 28covariant constant, 64covariant derivative, 64Crofton’s formula, 24, 32cross ratio, 93cubiccuspidal, 2nodal, 2twisted, 3curvature, 11cycloid, 3Darboux frame, 67, 98, 99developable, see ruled surface, developabledirectrix, 37Dupin indicatrix, 53eigenvalue, 108eigenvector, 108elliptic point, 48Euler characteristic, 81exterior angle, 31, 79first fundamental form, 38flat, 58, 73, 80, 86, 98Frenet formulas, 11Frenet frame, 10functional, 101Gauss equation, 57, 60, 98Gauss map, 24, 43Gauss-Bonnet formula, 79, 82, 90, 99Gauss-Bonnet Theorem, 90global, 81local, 79Gaussian curvature, 48, 51, 54, 57, 78, 98constant, 59, 87generalized helix, 15geodesic, 67geodesic curvature, 68globally isometric, 71gradient, 109Gronwall inequality, 113H, 48helicoid, 35, 47, 52, 63helix, 3holonomy, 75, 78horocycle, 92hyperbolic plane, 87Klein-Beltrami model, 95Poincaré model, 94hyperbolic point, 48inversion, 93involute, 19117
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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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8 Chapter 1. CurvesAn important obs
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10 Chapter 1. Curvesof the road, st
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12 Chapter 1. CurvesSummarizing,κ(
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14 Chapter 1. Curvescurvature κ. I
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16 Chapter 1. CurvesFigure 2.2Conve
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18 Chapter 1. Curvesalso Exercise 2
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20 Chapter 1. Curvesto the curve β
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22 Chapter 1. Curvesthe osculating
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24 Chapter 1. CurvesWe turn now to
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26 Chapter 1. Curvesintersect C eit
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28 Chapter 1. Curvesh(s,t)α(t)α(s
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30 Chapter 1. Curvesy( x(s) ,y(s))
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32 Chapter 1. Curvesɛ2ɛ 1ɛ 3CFig
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CHAPTER 2Surfaces: Local Theory1. P
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36 Chapter 2. Surfaces: Local Theor
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Page 110 and 111: 108 Appendix. Review of Linear Alge
Page 116 and 117: ANSWERS TO SELECTED EXERCISES1.1.1
Page 120: 118 INDEXisometry, 107K, 48k-point
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DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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