Introduction and List of Contents
I have included below a few of the reviews of this book.
NICOLAESCU, Liviu I.
Lectures on the geometry of manifolds. (English)
World Scientific Publishing Co., Inc., River Edge, NJ,
1996. xviii+481 pp. $62.00. ISBN 981-02-2836-8
This 450-page book takes the reader from the first principles of smooth manifold theory to advanced analysis on manifolds involving the solution of both linear and nonlinear elliptic equations. The author writes: "In writing this book we had in mind the beginning graduate student who wants to specialize in global geometric analysis in general and gauge theory in particular." He has succeeded in producing a volume that will be of great value to such a student, both as a course text and as a reference.
A brief outline shows that the book covers many standard topics which such a student should be expected to know. Manifolds, the bundles associated to the tangent bundle, various kinds of derivative, and integration appear in Chapters 1--3. Chapter 4 deals with the elements of Riemannian geometry and Chapter 5 with the calculus of variations, particularly as applied to geodesics. In Chapter 6 we find discussion of the fundamental group; for example, Myers' theorem is used to show that a positively Ricci-curved manifold has finite fundamental group. The long Chapter 7 deals (from scratch) with de Rham cohomology and (briefly) with the Cech cohomology of sheaves. In Chapter 8 characteristic classes are approached using the Chern-Weil construction. Chapter 9 deals with elliptic equations on manifolds. With an eye to nonlinear problems ahead, the $L\sp p$ and $C\sp \alpha$ theories are developed, rather than simply the $L\sp 2$ theory. In Chapter 10 the Dirac operators associated to a bundle of Clifford modules are considered as examples.
The book would have benefitted from the services of a careful editor. An errata sheet is included, but quite a few further errata remain for the reader to find. A sentence such as "The exterior derivative also has a fibered version by its true meaning can be grasped by refering [sic] to Leray's spectral sequence of a fibration and so we will not deal with it", where a typo and the author's sometimes unconventional English interact, is not likely to be comprehensible to the novice reader. I checked, at random, four page references from the index; they were all incorrect. These things matter.
I have left until last the greatest virtue of the book, which is its presentation of a large number of interesting, significant, and up-to-the-minute examples. "Supersymmetric" terminology is used throughout the book, whenever it is of value. The Poincare polynomials of Grassmannians are computed, using de Rham cohomology, by means of invariant integration on Lie groups. This gives the author an opportunity for a little digression (there are a number of these throughout the book) on Young tableaux and Schur functions. The nonlinear elliptic theory is illustrated by a discussion of the Kazdan-Warner problem of prescribing curvatures; this leads to a proof of the uniformization theorem. The final section of the book introduces $\roman{Spin}\sp c$ and the associated Dirac operator. All these are very valuable discussions and are difficult to find elsewhere in a book at this level. In all, this would be an excellent text and reference for an introductory course or series of courses in this active area of mathematics.
© Copyright American Mathematical Society 1997, 1998
This book is a recommendable exposition of differential
geometry. \par The text begins with preparatory material, such as
the notion of smooth manifolds, vector bundles, and aspects of
tensor calculus. Chapter 1 introduces the concept of smooth
manifolds and the notion of Lie groups. Chapter 2 deals with
basic constructions on manifolds. Chapter 3 contains techniques
of the calculus on manifolds, the Lie derivative, connections on
vectors bundles, integration on manifolds as well as a whole
section on representation theory of compact Lie groups. Chapter 4
deals with Riemannian geometry including the geometry of
submanifolds and concludes with the Gauss-Bonnet theorem. A brief
chapter containing basic techniques of the calculus of variations
(Chapter 5) is followed by a discussion of the fundamental group
and covering spaces (Chapter 6). Chapter 7 presents topological
material describing DeRham cohomology, Poincare duality,
intersection theory, symmetric spaces, and Cech cohomology,
whereas Chapter 8 introduces the tool of characteristic classes.
The topic of Chapter 9 is the study of elliptic equations on
manifolds starting with basic notions on partial differential
operators and concluding with Fredholm, spectral, and Hodge
theory. The book concludes with an introductory chapter on Dirac
operators including fundamental examples and the $\text{spin}\sp
c$ case. \par The book is marked by its clear presentation,
contains many exercises and is illustrated by numerous detailed
examples.
[ K.Habermann (Leipzig) , Zentralbaltt]