Differential Geometry: Manifolds Curves and Surfaces

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.