Preface
This book is an introductory graduate-level textbook on the theory of
smooth manifolds, for students who already have a solid acquaintance with
general topology, the fundamental group, and covering spaces, as well as
basic undergraduate linear algebra and real analysis. It is a natural sequel
to my earlier book on topological manifolds [Lee00].
This subject is often called “differential geometry.” I have mostly avoided
this term, however, because it applies more properly to the study of smooth
manifolds endowed with some extra structure, such as a Riemannian met-
ric, a symplectic structure, a Lie group structure, or a foliation, and of the
properties that are invariant under maps that preserve the structure. Al-
though I do treat all of these subjects in this book, they are treated more as
interesting examples to which to apply the general theory than as objects
of study in their own right. A student who finishes this book should be
well prepared to go on to study any of these specialized subjects in much
greater depth.
The book is organized roughly as follows. Chapters 1 through 4 are
mainly definitions. It is the bane of this subject that there are so many
definitions that must be piled on top of one another before anything in-
teresting can be said, much less proved. I have tried, nonetheless, to bring
in significant applications as early and as often as possible. The first one
comes at the end of Chapter 4, where I show how to generalize the classical
theory of line integrals to manifolds.
The next three chapters, 5 through 7, present the first of four major
foundational theorems on which all of smooth manifolds theory rests—the
inverse function theorem—and some applications of it: to submanifold the-
