From the standpoint of non-experts needing to learn some differential geometry for use in other parts of mathematics the choice of text is often a bit of a problem. Spivak’s *Calculus on Manifolds* comes to mind, possibly, as a place to start, but where to go from there? Well, Spivak’s titanic *A Comprehensive Introduction to Differential Geometry* is always there, of course, but this famous series of five(!) volumes seems to be gauged toward geometers *per se*; moreover, at this level we also have such works as *Algebraic Topology via Differential Topology*, by Karoubi and Leruste, or Warner’s *Foundations of Differential Geometry and Lie Groups*, or Boothby’s *An Introduction to Differentiable Manifolds and Riemannian Geometry* to consider.

To be sure, these fine books, and many others besides, are aimed at graduate students and have made their mark in no uncertain terms. But it cannot be denied that the plethora of connections between differential geometry, or the theory of differentiable manifolds, and other parts of mathematics (algebraic topology, Lie groups, Riemannian geometry, etc.) makes for some bewilderment of the part of the student or reader. On top of this the whole business is complicated (complexified?) by recent breakthroughs in mathematics leading to the solutions of famous problems, from Calabi-Yau to Poincaré *à la* Perelman: so many irresistible novel directions.

This having been said, it is obviously a consummation devoutly to be wished to have available a beginning text whose goal it is to initiate a continuous trajectory from what a first year graduate student should know to advanced material situated at the intersection of different areas of high current interest. Tu’s *An Introduction to Manifolds* is accordingly offered as the first of a quartet of works that should make for a fine education in differential geometry in this broader modern sense, tailored to many subsequent applications. The other three books in the sequence are the already well-established *Differential Forms in Algebraic Topology* , written together with Tu’s mentor, the late Raoul Bott (1926–2005), Tu’s evidently forthcoming *Differential Geometry: Connections, Curvature, and Characteristic Classes*, and, finally, another joint project with Bott, *Elements of Equivariant Cohomology* (due in a year). The centrality of algebraic topology in this sequence is particularly noteworthy: it is eminently appropriate, given what has transpired over the last fifty years or so.

Turning to *An Introduction to Manifolds* , the book under review, Tu notes explicitly that his objective is to provide “a readable but rigorous introduction that gets the reader quickly up to speed, to the point where he or she can compute de Rham cohomology of simple spaces.” Tu also opts to “make the first four chapters of the book independent of point-set topology,” leaving the latter to an Appendix (which the reader should consult, however). This certainly aids in getting the reader “quickly up to speed,” and, indeed, makes for a smooth presentation. This holds for the whole book, in fact.

*An Introduction to Manifolds* is split up into eight parts, well organized, well written, and, as Tu claims, readable. Their titles are, “Euclidean spaces,” “Manifolds,” “The tangent space,” “Lie groups and Lie algebras,” “Differential forms,” “Integration,” “De Rham theory,” and “Appendices.” Everything is in place to provide a truly first-rate foundation for what comes next.

This excellent and accessible book also comes equipped with plenty of examples and exercises, whence it will serve well as both a classroom text and a source for self-study. Indeed, I propose to use it myself, given that I am one of the non-experts mentioned in the first sentence of this review.

Michael Berg is Professor of Mathematics at Loyola Marymount University.

## Comments

## Andrew Locascio

I really don't have much to add to Dr. Berg's review except the perspective of one of the target audience for this wonderful book, i.e., of a graduate student attempting to learn modern manifold theory. I'm currently enrolled in the second semester of the first year graduate differential geometry course taught by the very respected geometer, Józef Dodziuk, who is also a very fine teacher. Since I couldn't take the first half of the course last semester, I have had to teach myself some of the material.

As Berg notes, there are a host of introductory sources for students to learn this material - Spivak's opus probably being the most famous, and deservedly so. But despite its beauty and the awesome historical perspective Spivak brings to his classic, time is not a graduate student's friend and - let's face it - it's just not practical to use as a textbook!

John M.Lee's

Smooth Manifolds- one of the references for my course - is another wonderfully written choice, but it also is very long, and it requires more background then Spivak. Boothby is a little sloppy and leaves out too many topics for my taste, although itiswell written. And S.S. Chern'sLecture in Differential Geometry, while beautiful and well worth reading, is much terser then the previous works, requires a lot more background and has themajordrawback of having no exercises.Tu's book is just what the doctor ordered. It is of managable length, it has very reasonable prequisites (a year of real analysis and a year of abstract algebra) and - best of all - it's got a lot of

well chosenpictures, also important for a geometry book. When it is finished, this tetraology may become the new gold standard for preparing students for their prelims and qualifiers.