DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces

More documents

Recommendations

Info

84 Chapter 3. Surfaces: Further TopicsIt is very interesting that the total curvature does not change as we deform the surface, for example,as shown in Figure 1.9. In a topology course, one proves that any closed, oriented surface without∫∫MKdA = 4πFigure 1.9boundary must have the topological type of a sphere or of a g-holed torus for some positive integerg. Thus (cf. Exercises 5 and 6), the possible Euler characteristics of such a surface are 2, 0, −2,−4, ...; moreover, the integral ∫∫ MKdA determines the topological type of the surface.We conclude this section with a few applications of the Gauss-Bonnet Theorem.Example 5. Suppose M is a surface of nonpositive Gaussian curvature. Then there cannot bea geodesic 2-gon R on M that bounds a simply connected region. For if there were, by Theorem1.5 we would have∫∫0 ≥ KdA =2π − (ɛ 1 + ɛ 2 ) > 0,Rwhich is a contradiction. (Note that the exterior angles must be strictly less than π because thereis a unique (smooth) geodesic with a given tangent direction.) ▽Example 6. Suppose M is topologically equivalent to a cylinder and its Gaussian curvatureis negative. Then there is at most one simple closed geodesic in M. Note, first, as indicated inRα must be like one of theseFigure 1.10Figure 1.10, that if there is a simple closed geodesic α, itmust separate M (i.e., does not bounda region) or else it bounds a disk R, inwhich case we would have 0 > ∫∫ RKdA =2πχ(R) =2π,which is a contradiction. On the other hand, suppose there were two. If they don’t intersect, thenthey bound a cylinder R and we get 0 > ∫∫ RKdA =2πχ(R) =0,which is a contradiction. If theydo intersect, then we we have a geodesic 2-gon bounding a simply connected region, which cannothappen by Example 5. ▽
§1. Holonomy and the Gauss-Bonnet Theorem 85EXERCISES 3.11. Compute the holonomy around a parallel on*a. a torusb. the paraboloid x(u, v) =(u cos v, u sin v, u 2 )c. the catenoid x(u, v) =(cosh u cos v, cosh u sin v, u)2. Determine whether there can be a (smooth) closed geodesic on a surface whena. K>0b. K =0c. K0b. K =0c. K0 that is topologically a cylinder. Prove that there cannot betwo disjoint simple closed geodesics both going around the neck of the surface.9. Consider the paraboloid M parametrized by x(u, v) =(u cos v, u sin v, u 2 ), 0 ≤ u, 0≤ v ≤ 2π.Denote by M r that portion of the paraboloid defined by 0 ≤ u ≤ r. ∫a. Calculate the geodesic curvature of the boundary circle and compute κ g ds.∂M rb. Calculate χ(M r ).∫∫c. Use the Gauss-Bonnet Theorem to compute KdA. Find the limit as r →∞. (ThisM ris the total curvature of the paraboloid.)∫∫d. Calculate K directly (however you wish) and compute KdA explicitly.e. Explain the relation between the total curvature and the image of the Gauss map of M.10. Consider the pseudosphere (with boundary) M parametrized as in Example 6 of Chapter 2,Section 2, but here we take u ≥ 0. Denote by M r that portion defined by 0 ≤ u ≤ r. (Notethat we are including the boundary circle u = 0.)M
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 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

You also want an ePaper? Increase the reach of your titles

Delete template?

Save as template?