Geometría Diferencial - Shifrin

DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces
Preliminary Version Fall, 2008

Theodore Shifrin University of Georgia

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

c 2008 Theodore Shifrin
No portion of this work may be reproduced in any form without written permission of the author.

CONTENTS

1. CURVES

. . . . . . . . . . . . . . . . . . . .. . . . .
1

1

1. Examples, Arclength Parametrization 2. Local Theory: Frenet Frame 10 3. Some Global Results 22

2. SURFACES: LOCAL THEORY . . . . . . . . . . . . . . . . .
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 55 4. CovariantDifferentiation, Parallel Translation, and Geodesics 64

35

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

. . . . . . . . . . . . . . . .

76

Holonomy and the Gauss-Bonnet Theorem 76 An Introduction to Hyperbolic Geometry 88 Surface Theory with Differential Forms 97 Calculus of Variations and Surfaces of Constant Mean Curvature

102

Appendix. REVIEW OF LINEAR ALGEBRA AND CALCULUS
1. LinearAlgebra Review 109 2. Calculus Review 111 3. Differential Equations 114

. . . . . .

109

SOLUTIONS TO SELECTED EXERCISES

. . . . . . . . . . .

116 119

INDEX . . . . . . . . . . . . . . . . . . . . . . . . .

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 referenceis made later) are marked with a sharp (♯ ). Fall, 2008

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 ′′ , . . . , 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 α : I → R3 for some intervalI = (a, b) or [a, b] in R (possibly infinite). We say α is regular if α′ (t) = 0 for all t ∈ I. We can imagine a particle moving along the path α, with its position at time t given by α(t). As we learned in vector calculus, α′ (t) = dα α(t + h) − α(t) = lim h→0 dt h

is the velocity of the particle at time t. The velocity vector α′ (t) is tangent to the curve at α(t) and its length, α′ (t) , isthe speed of the particle. 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 = P Q = Q − P and set α(t) = P + tv, t ∈ R. Note that α(0) = P , α(1) = Q, and for 0 ≤ t ≤ 1, α(t) is on the line segment P Q. We ask the reader to check in Exercise 8 that of all paths from P to Q, the“straight line path” α 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): α(t) = a cos t, sin t = a cos t, a sin t , 0 ≤ t ≤ 2π.

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

(a)
Figure1.1
1

(b)

2

Chapter 1. Curves

(c) Now, if a, b > 0 and we apply the linear map T : R2 → R2 , T (x, y) = (ax, by),

we see that the unit circle x2 + y 2 = 1 maps to the ellipse x2 /a2 + y 2 /b2 = 1. Since T (cos t, sin t) = (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 inFigure 1.2. On the left is the cuspidal

y=tx

y2=x3+x2 y2=x3

(a)

(b)

Figure 1.2 cubic y 2 = x3 , and on the right is the nodal cubic y 2 = x3 + x2 . These can be parametrized, respectively, by the functions α(t) = (t2 , t3 ) and α(t) = (t2 − 1, t(t2 − 1)).

(In the latter case, as the figure suggests, we see that the line y = tx intersects the curve when (tx)2 = x2 (x + 1), so...