DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces
Preliminary Version Fall, 2015
Theodore Shifrin University of Georgia
Dedicated to the memory of Shiing-Shen Chern, my adviser and friend
c 2015 Theodore Shifrin
No portion of this work may be reproduced in any form without written permission of 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. SomeGlobalResults 23
2. SURFACES: LOCAL THEORY ...... 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: FURTHER TOPICS ...... 79 1. Holonomy and the Gauss-Bonnet Theorem 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. REVIEW OF LINEAR ALGEBRA AND CALCULUS . . . 114 1. Linear Algebra Review 114 2. Calculus Review 116 3. Differential Equations 118
SOLUTIONS TO SELECTED EXERCISES ...... 121
INDEX ...... 124
Problems to which answers or hints are given at the back of the book are marked with an asterisk (*). Fundamental exercises that are particularly important (and to which reference is made later) are marked with a sharp (]).
August, 2015 CHAPTER 1 Curves
1. Examples, Arclength Parametrization
We say a vector function f .a; b/ R3 is Ck (k 0; 1; 2; : : :) if f and its first k derivatives, f , f ,..., W ! D 0 00 f.k/, exist and are all continuous. We say f is smooth if f is Ck for every positive integer k.A parametrized curve is a C3 (or smooth) map ˛ I R3 for some interval I .a; b/ or Œa; b in R (possibly infinite). We W ! D say ˛ is regular if ˛ .t/ 0 for all t I . 0 ¤ 2 We can imagine a particle moving along the path ˛, with its position at time t given by ˛.t/. As we learned in vector calculus, d˛ ˛.t h/ ˛.t/ ˛0.t/ lim C D dt D h 0 h ! is the velocity of the particle at time t. The velocity vector ˛0.t/ is tangent to the curve at ˛.t/ and its length, ˛ .t/ , is the speed of the particle. k 0 k Example 1. We begin with some standard examples. (a) Familiar from linear algebra and vector calculus is a parametrized line: Given points P and Q in R3, we let v !PQ Q P and set ˛.t/ P tv, t R. Note that ˛.0/ P , ˛.1/ Q, D D D C 2 D D 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/ a cos t; sin t a cos t; a sin t ; 0 t 2: D D
(a cos t, a sin t) (a cos t, b sin t) b t a a
(a) (b)
FGUREI 1.1
1 2 CHAPTER 1. CURVES
(c) Now, if a; b > 0 and we apply the linear map
T R2 R2; T .x; y/ .ax; by/; W ! D we see that the unit circle x2 y2 1 maps to the ellipse x2=a2 y2=b2 1. Since T.cos t; sin t/ C D C D D .a cos t; b sin t/, 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
y2=x3+x2
y2=x3
(a) (b)
GUREFI 1.2
y2 x3, and on the right is the nodal cubic y2 x3 x2. These can be parametrized, respectively, D D C by the functions
˛.t/ .t2; t3/ and ˛.t/ .t2 1; t.t2 1//: D D (In the latter case, as the figure suggests, we see that the line y tx intersects the curve when D .tx/2 x2.x 1/, so x 0 or x t2 1.) D C D D
z2=y3 z=x3
y=x2
FGUREI 1.3 .E÷1 XAMPLES,ARCLENGTH PARAMETRIZATION 3
(e) Now consider the twisted cubic in R3, illustrated in Figure 1.3, given by ˛.t/ .t; t2; t3/; t R: D 2 Its projections in the xy-, xz-, and yz-coordinate planes are, respectively, y x2, z x3, and D D z2 y3 (the cuspidal cubic). D (f) Our next example is a classic called the cycloid: It is the trajectory of a dot on a rolling wheel (circle). Consider the illustration in Figure 1.4. Assuming the wheel rolls without slipping, the
P t a O
FGUREI 1.4
distance it travels along the ground is equal to the length of the circular arc subtended by the angle through which it has turned. That is, if the radius of the circle is a and it has turned through angle t, then the point of contact with the x-axis, Q, is at units to the right. The vector from the origin to
C C a P t a cos t P a sin t O Q
FGUREI 1.5
the point P can be expressed as the sum of the three vectors !OQ, !QC , and !CP (see Figure 1.5):