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DIFFERENTIAL GEOMETRY: A First Course in Curves and …

DIFFERENTIAL geometry : A First Course in Curves and Surfaces Preliminary Version Summer, 2016. Theodore Shifrin University of Georgia Dedicated to the memory of Shiing-Shen Chern, my adviser and friend 2016. c 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. 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 Gauss-Bonnet Theorem 79. 2. An Introduction to Hyperbolic geometry 91.

DIFFERENTIAL GEOMETRY: A First Course in Curves and Surfaces Preliminary Version Summer, 2016 Theodore Shifrin University of Georgia Dedicated to the memory of Shiing-Shen Chern,

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