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# dg - Elementary Dierential Geometry Lecture Notes Gilbert...

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Elementary Differential Geometry: Lecture Notes Gilbert Weinstein

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Contents Preface 5 Chapter 1. Curves 7 1. Preliminaries 7 2. Local Theory for Curves in R 3 8 3. Plane Curves 10 4. Fenchel’s Theorem 14 Exercises 16 Chapter 2. Local Surface Theory 19 1. Surfaces 19 2. The First Fundamental Form 21 3. The Second Fundamental Form 23 4. Examples 25 5. Lines of Curvature 28 6. More Examples 30 7. Surface Area 33 8. Bernstein’s Theorem 37 9. Theorema Egregium 39 Exercises 41 Chapter 3. Local Intrinsic Geometry of Surfaces 45 1. Riemannian Surfaces 45 2. Lie Derivative 47 3. Covariant Differentiation 48 4. Geodesics 50 5. The Riemann Curvature Tensor 53 6. The Second Variation of Arclength 56 Exercises 59 Index 61 3

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Preface These notes are for a beginning graduate level course in differential geometry. It is assumed that this is the students’ first course in the subject. Thus the choice of subjects and presentation has been made to facilitate as much as possible a concrete picture. For those interested in a deeper study, a second course would take a more abstract point of view, and in particular, could go further into Riemannian geometry. Much of the material is borrowed from the following sources, but has been adapted according to my own taste: [1] M. P. Do Carmo , Differential geometry of curves and surfaces , Prentice-Hall. [2] L. P. Eisenhart An introduction to differential geometry with use of the ten- sor calculus , Princeton University Press. [3] W. Klingenberg , A course in differential geometry , Springer-Verlag. [4] B. O’Neill Elementary differential geometry , Academic Press. [5] M. Spivak , A comprehensive introduction to Differential Geometry , Publish or Perish. [6] J. J. Stoker , Differential Geometry , Wiley & Sons. The prerequisites for this course are: linear algebra , preferably with some ex- posure to multilinear algebra; calculus up to and including the inverse and implicit function theorem; the fundamental theorem of ordinary differential equations con- cerning existence of solutions, uniqueness, and continuous dependence on parame- ters, and some knowledge of linear systems of ordinary differential equations; linear first order partial differential equations ; complex analysis including Liouville’s the- orem; and some elementary topology . It is highly recommended for the students to complete all the exercises included in these notes. Gilbert Weinstein Birmingham, Alabama April 2000 5

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CHAPTER 1 Curves 1. Preliminaries Definition 1.1 . A parametrized curve is a smooth ( C ) function γ : I R n . A curve is regular if γ = 0. When the interval I is closed, we say that γ is C on I if there is an interval J and a C function β on J which agrees with γ on I . Definition 1.2 . Let γ : I R n be a parametrized curve, and let β : J R n be another parametrized curve. We say that β is a reparametrization (orientation- preserving reparametrization) of γ if there is a smooth map τ : J I with τ > 0 such that β = γ τ .

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dg - Elementary Dierential Geometry Lecture Notes Gilbert...

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