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