DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

More documents

Recommendations

Info

62 Chapter 2. Surfaces: Local Theory2. Use the first Gauss equation to derive the formula (∗) given on p. 57 for Gaussian curvature.3. Check the Gaussian curvature of the sphere using the formula (∗) onp.57.4. Check that for a parametrized surface with E = G = λ(u, v) and F =0,the Gaussian curvatureis given by K = − 12λ ∇2 (ln λ). (Here ∇ 2 f = ∂2 f∂u 2 + ∂2 fis the Laplacian of f.)∂v2 5. Calculate the Christoffel symbols for the following parametrized surfaces. Then check in eachcase that the Codazzi equations and the first Gauss equation hold.a. the plane, parametrized by polar coordinates: x(u, v) =(u cos v, u sin v, 0)b. a helicoid: x(u, v) =(u cos v, u sin v, v)♯ c. a cone: x(u, v) =(u cos v, u sin v, cu), c ≠0♯ *d. a surface of revolution: x(u, v) = ( f(u) cos v, f(u) sin v, g(u) ) , with f ′ (u) 2 + g ′ (u) 2 =16. Prove there is no compact minimal surface M ⊂ R 3 .7. Decide whether there is a parametrized surface x(u, v) witha. E = G =1,F =0,l =1=−n, m =0b. E = G =1,F =0,l = e u = n, m =0c. E =1,F =0,G = cos 2 u, l = cos 2 u, m =0,n =18. a. Modify the proof of Theorem 3.5 to prove that a smooth, compact surface with K>0 andconstant mean curvature is a sphere.b. Give an example to show that the result of Lemma 3.6 fails if we assume k 1 has a localminimum and k 2 has a local maximum at P .9. Give examples of (locally) non-congruent parametrized surfaces x and x ∗ witha. I=I ∗b. II = II ∗ (Hint: Try reparametrizing some of our simplest surfaces.)10. Let x(u, v) =α(u) +vβ(u) beaparametrization of a ruled surface. Prove that the tangentplane is constant along rulings (i.e., the surface is flat) if and only if α ′ (u), β(u), and β ′ (u) arelinearly dependent for every u. (Hint: When is S(x v )=0? Alternatively, consider x u × x v andapply Exercise A.2.1.)11. Prove that α is a line of curvature in M if and only if the ruled surface formed by the surfacenormals along α is flat. (Hint: See Exercise 10.)12. Show that the Gaussian curvature of the parametrized surfacesx(u, v) =(u cos v, u sin v, v)y(u, v) =(u cos v, u sin v, ln u)is the same for each (u, v), and yet the first fundamental forms I x and I y do not agree. (Thus,the converse of Corollary 3.2 is false.)13. Suppose that the surface M is doubly ruled by orthogonal lines (i.e., through each point of Mthere pass two orthogonal lines).
§4. Covariant Differentiation, Parallel Translation, and Geodesics 63a. Using the Gauss equations, prove that K =0.b. Now deduce that M must be a plane.(Hint: As usual, assume the families of lines are u- and v-curves.)14. Suppose M is a surface with no umbilic points and one constant principal curvature k 1 ≠0.Prove that M is (a subset of) a tube of radius r =1/|k 1 | about a curve. That is, there is acurve α so that M is (a subset of) the union of circles of radius r in each normal plane, centeredalong the curve. (Hints: As usual, work with a parametrization where the u-curves are linesof curvature with principal curvature k 1 and the v-curves are lines of curvature with principalcurvature k 2 . Use the Codazzi equations to show that the u-curves have curvature |k 1 | and areplanar. Then define α appropriately and check that it is a regular curve.)15. Consider the parametrized surfacesx(u, v) =(− cosh u sin v, cosh u cos v, u)y(u, v) =(u cos v, u sin v, v)(a helicoid).(a catenoid)a. Compute the first and second fundamental forms of both surfaces, and check that bothsurfaces are minimal.b. Find the asymptotic curves on both surfaces.c. Show that we can locally reparametrize the helicoid in such a way as to make the firstfundamental forms of the two surfaces agree; this means that the two surfaces are locallyisometric. (Hint: See p. 39. Replace u with sinh u in the parametrization of the helicoid.Why is this legitimate?)d. Why are they not globally isometric?e. (for the student who’s seen a bit of complex variables) As a hint to what’s going on here,let z = u + iv and Z = x + iy, and check that, continuing to use the substitution frompart c, Z = (sin iz, cos iz, z). Understand now how one can obtain a one-parameter familyof isometric surfaces interpolating between the helicoid and the catenoid.16. Find all the surfaces of revolution of constant curvaturea. K =0b. K =1c. K = −1(Hint: There are more than you might suspect. But your answers will involve integrals youcannot express in terms of elementary functions.)4. Covariant Differentiation, Parallel Translation, and GeodesicsNow we turn to the “intrinsic” geometry of a surface, i.e., the geometry that can be observed byan inhabitant (for example, a flat ant) of the surface, who can only perceive what happens along (or,say, tangential to) the surface. Anyone who has studied Euclidean geometry knows how importantthe notion of parallelism is (and classical non-Euclidean geometry arises when one removes Euclid’s
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
Page 12 and 13:
10 Chapter 1. Curvesof the road, st
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
Page 38 and 39: 36 Chapter 2. Surfaces: Local Theor
Page 78 and 79: 76 Chapter 3. Surfaces: Further Top
Page 102 and 103: 100 Chapter 3. Surfaces: Further To
Page 110 and 111: 108 Appendix. Review of Linear Alge
Page 112 and 113: 110 Appendix. Review of Linear Alge
Page 114 and 115:
112 Appendix. Review of Linear Alge
Page 116 and 117:
ANSWERS TO SELECTED EXERCISES1.1.1
Page 118 and 119:
116 SELECTED ANSWERS2.4.8 Only circ
Page 120:
118 INDEXisometry, 107K, 48k-point
show all

DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

Create successful ePaper yourself

Delete template?

Save as template?