DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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72 Chapter 2. Surfaces: Local Theory2. Prove that κ 2 = κ 2 g + κ 2 n.3. Suppose α is a non-arclength-parametrized curve. Using the formula (∗∗) onp.14, prove thatthe velocity vector of α is parallel along α if and only if κ g =0and υ ′ =0.*4. Find the geodesic curvature κ g of a latitude circle u = u 0 on the unit spherea. directlyb. by applying the result of Exercise 25. Check that the parallel u = u 0 is a geodesic on the surface of revolution parametrized as inProposition 4.4 if and only if f ′ (u 0 )=0. Give a geometric interpretation of and explanationfor this result.6. Use the equations (♣) todetermine through what angle a vector turns when it is paralleltranslatedonce around the circle u = u 0 on the cone x(u, v) =(u cos v, u sin v, cu), c ≠0. (SeeExercise 2.3.5c.)7. a. Prove that if the surfaces M and M ∗ are tangent along the curve C, parallel translationalong C is the same in both surfaces.b. Use the result of part a to determine the effect of parallel translation around the latitudecircle u = u 0 on the unit sphere. Figure 4.6 and basic Euclidean geometry may be of help.(Cf. Example 3.)Figure 4.6*8. What curves lying on a sphere have constant geodesic curvature?9. Use the equations (♣♣) tofind the geodesics on the plane parametrized by polar coordinates.(Hint: Examine Example 6(b).)♯ 10. a. Suppose F =0and the u-curves are geodesics. Use the equations (♣♣) toprove that Eis a function of u only.b. Suppose F =0and the u- and v-curves are geodesics. Prove that the surface is flat.11. a. Prove that an arclength-parametrized curve α on a surface M with κ ≠0is a geodesic ifand only if n = ±N.b. Let α be a space curve, and let M be the ruled surface generated by its binormals. Provethat the curve is a geodesic on M.
§4. Covariant Differentiation, Parallel Translation, and Geodesics 7312. a. Prove that if a geodesic is planar and not a line, then it is a line of curvature. (Hint: UseExercise 11.)b. Prove that if every geodesic of a (connected) surface is planar, then the surface is containedin a plane or a sphere.13. Prove or give a counterexample:a. A line lying in a surface is both an asymptotic curve and a geodesic.b. If a curve is both an asymptotic curve and a geodesic, then it must be a line.c. If a curve is both a geodesic and a line of curvature, then it must be planar.14. Show that the geodesic curvature at P of a curve C in M is equal (in absolute value) to thecurvature at P of the projection of C into T P M.15. Check using Clairaut’s relation, Proposition 4.4, that great circles are geodesics on a sphere.(Hint: The result of Exercise A.1.3 may be useful.)16. Let M be asurface and P ∈ M. WesayU, V ∈ T P M are conjugate if II P (U, V) =0.a. Let C ⊂ M be a curve. Define the envelope M ∗ of the tangent planes to M along C tobe the ruled surface whose generator at P ∈ C is the limiting position as Q → P of theintersection line of the tangent planes to M at P and Q. Prove that the generator at P isconjugate to the tangent line to C at P .b. Prove that if C is nowhere tangent to an asymptotic direction, then M ∗ is smooth (at leastnear C). Prove, moreover, that M ∗ is tangent to M along C and is a developable (flatruled) surface.c. Apply part b to give a geometric way of computing parallel translation. In particular, dothis for a latitude circle on the sphere. (Cf. Exercise 7.)17. Suppose that on a surface M the parallel translation of a vector from one point to anotheris independent of the path chosen. Prove that M must be flat. (Hint: Fix an orthonormalbasis e o 1 , eo 2 for T P M and define vector fields e 1 , e 2 by parallel translating. Choose coordinatesso that the u-curves are always tangent to e 1 and the v-curves are always tangent to e 2 . SeeExercise 10.)18. Use the Clairaut relation, Proposition 4.4, to describe the geodesics on the torus as parametrizedin Example 1(c) of Section 1. (Start with a geodesic starting at and making angle φ 0 withthe outer parallel. Your description should distinguish between the cases cos φ 0 ≤ a−ba+b andcos φ 0 > a−ba+b. Which geodesics never cross the outer parallel at all? Also, remember thatthrough each point there is a unique geodesic in each direction.)19. Use the proof of the Clairaut relation, Proposition 4.4, to show that a geodesic on a surface ofrevolution is given in terms of the standard parametrization in Example 7 of Section 2 by∫v = cduf(u) √ f(u) 2 − c 2 + const.
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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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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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Page 102 and 103: 100 Chapter 3. Surfaces: Further To
Page 110 and 111: 108 Appendix. Review of Linear Alge
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Page 118 and 119: 116 SELECTED ANSWERS2.4.8 Only circ
Page 120: 118 INDEXisometry, 107K, 48k-point
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DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

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