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.