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Unformatted text preview: DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces Preliminary Version Fall, 2011 Theodore Shifrin University of Georgia Dedicated to the memory of ShiingShen 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 1. Examples, Arclength Parametrization 1 2. Local Theory: Frenet Frame 10 3. Some Global Results 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 GaussBonnet 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 ( ] ). April, 2011 CHAPTER 1 Curves 1. Examples, Arclength Parametrization We say a vector function f W .a;b/ ! R 3 is C k ( k D 0;1;2;::: ) if f and its first k derivatives, f , f 00 , . . . , f .k/ , are all continuous. We say f is smooth if f is C k for every positive integer k . A parametrized curve is a C 3 (or smooth) map W I ! R 3 for some interval I D .a;b/ or a;b in R (possibly infinite). We say is regular if .t/ 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, .t/ D d dt D lim h ! .t C h/ NUL .t/ h is the velocity of the particle at time t . The velocity vector .t/ is tangent to the curve at .t/ and its length, k .t/ k , is the 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 R 3 , we let v D NULNUL! PQ D Q NUL P and set .t/ D P C t v , t 2 R . Note that .0/ D P , .1/ D Q , and for 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....
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