Differential Geometry: Manifolds, Curves, and Surfaces

Marcel Berger, Bernard Gostiaux (auth.)0387966269, 9780387966267

This book consists of two parts, different in form but similar in spirit. The first, which comprises chapters 0 through 9, is a revised and somewhat enlarged version of the 1972 book Geometrie Differentielle. The second part, chapters 10 and 11, is an attempt to remedy the notorious absence in the original book of any treatment of surfaces in three-space, an omission all the more unforgivable in that surfaces are some of the most common geometrical objects, not only in mathematics but in many branches of physics. Geometrie Differentielle was based on a course I taught in Paris in 1969- 70 and again in 1970-71. In designing this course I was decisively influ­ enced by a conversation with Serge Lang, and I let myself be guided by three general ideas. First, to avoid making the statement and proof of Stokes’ formula the climax of the course and running out of time before any of its applications could be discussed. Second, to illustrate each new notion with non-trivial examples, as soon as possible after its introduc­ tion. And finally, to familiarize geometry-oriented students with analysis and analysis-oriented students with geometry, at least in what concerns manifolds.

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Table of contents :
Front Matter….Pages i-xii
Background….Pages 1-29
Differential Equations….Pages 30-46
Differentiable Manifolds….Pages 47-102
Partitions of Unity, Densities and Curves….Pages 103-127
Critical Points….Pages 128-145
Differential Forms….Pages 146-187
Integration of Differential Forms….Pages 188-243
Degree Theory….Pages 244-276
Curves: The Local Theory….Pages 277-311
Plane Curves: The Global Theory….Pages 312-345
A Brief Guide to the Local Theory of Surfaces in R 3 ….Pages 346-402
A Brief Guide to the Global Theory of Surfaces….Pages 403-441
Back Matter….Pages 443-476