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.
A Brief Introduction.- Part I. The Euclidean Space.- Smooth Functions on R(N).- Tangent Vectors In R(N) as Derivations.- Alternating K-Linear Functions.- Differential Forms on R(N).- Part II. Manifolds.- Manifolds.- Smooth Maps on A Manifold.- Quotient.- Part III. The Tangent Space.- The Tangent Space.- Submanifolds.- Categories And Functors.- The Image of A Smooth Map.- The Tangent Bundle.- Bump Functions and Partitions of Unity.- Vector Fields.- Part IV. Lie Groups and Lie Algebras.- Lie Groups.- Lie Algebras.- Part V. Differential Forms.- Differential 1-Forms.- Differential K-Forms.- The Exterior Derivative.- Part VI. Integration.- Orientations.- Manifolds With Boundary.- Integration on A Manifold.- Part VII. De Rham Theory.- De Rham Cohomology.- The Long Exact Sequence in Cohomology.- The Mayer-Vietoris Sequence.- Homotopy Invariance.- Computation of De Rham Cohomology.- Proof of Homotopy Invariance.- Appendix A. Point-Set Topology.- Appendix B. Inverse Function Theorem of R(N) And Related Results.- Appendix C. Existence of A Partition of Unity in General.- Appendix D. Solutions to Selected Exercises.- Bibliography.- Index.