Geo Diferencial

DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces
Preliminary Version Fall, 2011

Theodore Shifrin University of Georgia

Dedicated to the memory of Shiing-Shen Chern, my adviser and friend

c 2011 Theodore Shifrin
No portion of this work may be reproduced in any form without written permission of the author.

CONTENTS
1. CURVES . . . . . . . . . . . . . . . . . . . .
1.Examples, Arclength Parametrization 2. Local Theory: Frenet Frame 10 3. Some Global Results 23 1

1

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, ParallelTranslation, and Geodesics 66

3. SURFACES: FURTHER TOPICS
1. 2. 3. 4.

. . . . . . . . . . . 79

Holonomy and the Gauss-Bonnet Theorem 79 An Introduction to Hyperbolic Geometry 91 Surface Theory with Differential Forms 101 Calculus of Variations and Surfaces of Constant Mean Curvature

107

Appendix. REVIEW OF LINEAR ALGEBRA AND CALCULUS
1. Linear Algebra Review 114 2. Calculus Review116 3. Differential Equations 118

. . .

114

SOLUTIONS TO SELECTED EXERCISES . . . . . . . INDEX . . . . . . . . . . . . . . . . . . .

121 124

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 (] ). April, 2011CHAPTER 1 Curves
1. Examples, Arclength Parametrization We say a vector function fW .a; b/ ! R3 is Ck (k D 0; 1; 2; : : :) if f and its first k derivatives, f0 , f00 , . . . , f.k/ , 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 ˛W I ! R3 for some interval I D .a; b/ or Œa; b in R (possibly infinite). We say ˛ is regular if˛0 .t/ ¤ 0 for all t 2 I . 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 C h/ ˛.t/ ˛0 .t/ D D lim dt h h!0 is the velocity of the particle at time t. The velocity vector ˛0 .t/ is tangent to the curve at ˛.t/ and its length, k˛0 .t/k, is the speed of the particle. Example 1. We begin with some standardexamples. (a) Familiar from linear algebra and vector calculus is a parametrized line: Given points P and Q in ! R3 , 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 Ä t Ä 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 shallconsider 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 cos t; sin t D a cos t; a sin t ; 0Ät Ä2 :

(a cos t, a sin t) (a cos t, b sin t)

t a

b
a

(a)
F IGURE 1.1
1

(b)

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C HAPTER 1. C URVES

(c) Now, if a; b > 0 and weapply the linear map T W R2 ! R2 ; T .x; y/ D .ax; by/;

we see that the unit circle x 2 Cy 2 D 1 maps to the ellipse x 2 =a2 Cy 2 =b 2 D 1. Since T .cos t; sin t/ 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+x2y2=x3

(a)

(b)

F IGURE 1.2 y 2 D x 3 , and on the right is the nodal cubic y 2 D x 3 Cx 2 . These can be parametrized, respectively, by the functions ˛.t/ D .t 2 ; t 3 / and ˛.t/ D .t 2 1; t.t 2 1//:

(In the latter case, as the figure suggests, we see that the line y D tx intersects the curve when .tx/2 D x 2 .x C 1/, so x D 0 or x D t 2 1.)

z=x3

z2=y3

y=x2

F IGURE 1.3

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