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DIFFERENTIALGEOMETRY: AFirst Course in CurvesandSurfaces Preliminary Version Summer,2016 TheodoreShifrin University of Georgia Dedicated to the memory of Shiing-Shen Chern, myadviser and friend c 2016TheodoreShifrin Noportionofthis work may be reproducedin any form without written permissionof the author, other than duplication at nominal cost for those readers or students desiring a hard copy. CONTENTS 1. CURVES . . . . . . . . . . . . . . . . . . . . 1 1. Examples, Arclength Parametrization 1 2. Local Theory: Frenet Frame 10 3. Some Global Results 23 2. SURFACES:LOCALTHEORY . . . . . . . . . . . . 35 1. Parametrized Surfaces and the First Fundamental Form 35 2. The Gauss Map and the Second Fundamental Form 44 3. The Codazzi and Gauss Equations and the Fundamental Theorem of Surface Theory 57 4. Covariant Differentiation, Parallel Translation, and Geodesics 66 3. SURFACES:FURTHERTOPICS . . . . . . . . . . . 79 1. HolonomyandtheGauss-BonnetTheorem 79 2. An Introduction to Hyperbolic Geometry 91 3. Surface Theory with Differential Forms 101 4. Calculus of Variations and Surfaces of Constant Mean Curvature 107 Appendix. REVIEWOFLINEARALGEBRAANDCALCULUS . . . 114 1. Linear Algebra Review 114 2. Calculus Review 116 3. Differential Equations 118 SOLUTIONSTOSELECTEDEXERCISES . . . . . . . 121 INDEX . . . . . . . . . . . . . . . . . . . 124 Problemstowhichanswersorhintsaregivenatthebackofthebookaremarkedwith an asterisk (*). Fundamental exercises that are particularly important (and to which ] reference is made later) are marked with a sharp ( ). June, 2016 CHAPTER 1 Curves 1. Examples, Arclength Parametrization 3 k 0 00 Wesayavector function fW.a;b/ ! R is C (k D 0;1;2;:::) if f and its first k derivatives, f , f , ..., .k/ k f , exist and are all continuous. We say f is smooth if f is C for every positive integer k. A parametrized 3 3 curve is a C (or smooth) map ˛WI ! R for some interval I D .a;b/ or Œa;b in R (possibly infinite). We 0 say ˛ is regular if ˛ .t/ ¤ 0 for all t 2 I. Wecan imagine a particle moving along the path ˛, with its position at time t given by ˛.t/. As we learned in vector calculus, 0 d˛ ˛.t Ch/ ˛.t/ ˛.t/ D D lim dt h!0 h 0 is the velocity of the particle at time t. The velocity vector ˛ .t/ is tangent to the curve at ˛.t/ and its length, 0 k˛.t/k, is the speed of the particle. Example1. Webeginwithsomestandard examples. (a) Familiar from linear algebra and vector calculus is a parametrized line: Given points P and Q in 3 ! R , we let v D PQ D Q P and set ˛.t/ D P C tv, t 2 R. Note that ˛.0/ D P, ˛.1/ D Q, and for 0 t 1, ˛.t/ is on the line segment PQ. We ask the reader to check in Exercise 8 that of all paths from P to Q, the “straight line path” ˛ gives the shortest. This is typical of problems we shall consider in the future. (b) Essentially by the very definition of the trigonometric functions cos and sin, we obtain a very natural parametrization of a circle of radius a, as pictured in Figure 1.1(a): ˛.t/ D a cost;sint D acost;asint ; 0 t 2: (a cos t, a sin t) (a cos t, b sin t) t b a a (a) (b) FIGURE 1.1 1 2 CHAPTER1. CURVES (c) Now, if a;b > 0 and we apply the linear map TWR2 !R2; T.x;y/D.ax;by/; weseethattheunitcirclex2Cy2 D 1mapstotheellipsex2=a2Cy2=b2 D 1. SinceT.cost;sint/ D .acost;bsint/, the latter gives a natural parametrization of the ellipse, as shown in Figure 1.1(b). (d) Consider the two cubic curves in R2 illustrated in Figure 1.2. On the left is the cuspidal cubic y=tx 2 3 2 y =x +x 2 3 y =x (a) (b) FIGURE 1.2 y2 D x3,andontherightisthenodalcubicy2 D x3Cx2. Thesecanbeparametrized, respectively, bythe functions ˛.t/ D .t2;t3/ and ˛.t/ D .t2 1;t.t2 1//: (In the latter case, as the figure suggests, we see that the line y D tx intersects the curve when .tx/2 D x2.x C1/, so x D 0 or x D t2 1.) z2=y3 3 z=x y=x2 FIGURE 1.3
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