Customer Reviews

An Introduction to Differentiable Manifolds and Riemannian Geometry (Pure and Applied Mathematics, Volume 120)

5 star:		(4)
4 star:		(1)
3 star:		(1)
2 star:		(0)
1 star:		(1)

 

 

This book is an wonderful introduction to Differential Geometry for the serious student of mathematics. However, it is not aimed at engineers, physicists or even applied mathaticians.
The author assumes the reader has an extensive knowledge of abstract algebra and at least one course in analysis. Likewise, he has chosen to emphasis applications of the subject to Lie Groups, homotopy theory, and group actions, rather than the physical applications that applied mathematicians are looking for. But, for the student of pure mathematics, this text is a great starting point into the rich world of differential geometry.
Also, while this book is an introduction and requires no previous knowledge of the subject, it covers enough ground to be followed up by such topics as the Gauss-Bonnet Theorem, the Cartan-Hadamard Theorem, Bonnet's Theorem, or Morse Theory.

This book is a careful treatment of the subjects in the title. It is an introduction, but it manages to cover quite a bit of ground with lots of examples to illustrate. One of it's distinguishing points is the way in which the concrete, coordinate based calculations are emphasized even while usually presenting the more abstract, coordinate free approach as well.

The book does a good job at stimulating those studying it to develop intuition. I found the book helpful when I was first studying the subject.

My first course on manifolds was based on this book, and I believe that it is the best introduction to the subject (especially for beginners). I thoroughly enjoyed it! I should also recommend Conlon's 'Differentiable Manifolds' (2ed, Birkhauser), as it is the perfect follow-up to Boothby. --A

This book is masterfully written and excels for its clearness and elementary conception of every detail. It starts reviewing the necessary tools of analysis (inverse and implicit function theorems, constant rank theorem, existence and unicity of ordinary differential equations). Then, it dedicates much attention to motivate and construct the concept of a manifold M and the definition of the tangent space at a point of M (this is much harder to do for an abstract manifold than for a submanifold of the Euclidean space, and for the beginner, it demands a lot of training and time to master the different isomorphic disguises that the tangent space can adopt). Immediately, the book deals with submanifolds and submersions, vector fields and their one parameter flows, the Lie algebra of smooth vector fields and the Frobenius theorem. A very good introduction to Lie groups and Lie algebras follows, (including the correspondence between Lie subalgebras and Lie subgroups in any Lie group), discrete subgroups, the exponential map, the adjoint representation and homogeneous spaces. Later we get into integration and Stokes theorem, invariant integration on compact Lie groups (i.e.: Haar measure) and the Weyl decomposition theorem for representations of compact Lie groups. Fine, fine, fine. Next, Boothby introduce us in the realm of Riemannian geometry: covariant derivatives, parallel transport, the Levi-Civita connection, the Riemannian curvature, geodesics, normal neighbourhoods and of course the marvelous theorem of Hopf and Rinow. Maybe, here the pace is a bit faster: at places one needs pencil and paper to draw and compute. But overall, this chapter (the seventh) provides a rigourous and quick acquaintance with this vast part of geometry. A valuable glimpse on symmetric spaces ends this chapter. Finally, Boothby deals with some basic properties of curvature. First, he presents Gauss Theorema Egregium for surfaces in three dimensional Euclidean space. Then he gives Cartan structure equations for a Riemannian manifold, (using an arbitrary moving frame) and he proves that in a symmetric space the curvature tensor is parallel (Cartan's theorem). The book ends with Schur's theorem and manifolds of constant curvature. Most exercises are affordable. Maybe, an additional chapter is lacking, kind of a step further: Jacobi fields and cut loci, tubullar neighbourhoods and their volumes, Rauch comparison theorem... or maybe, further information on other geometric structures: complex, symplectic or contact structures (Boothby was a leading expert on contact geometry). But it would be unfair to forget that the author says that he wrote his book during a sabbatical year. Anyway, as it is, I swear I cannot find among 1000 introductory books one 10% better than this one, covering such a wide and realistic introduction to differential geometry, and demanding only such a little amount of prerequisites and so little effort to be read. Once you learn this book, you can go into Knapp's "Lie Groups: Beyond an Introduction" or Helgason's "Differential Geometry, Lie Groups, and Symmetric Spaces (Graduate Studies in Mathematics)" or Kobayashi and Nomizu's "Foundations of Differential Geometry (Wiley Classics Library) (Volume 1)", Sakai's "Riemannian Geometry (Translations of Mathematical Monographs)" Milnor's "Morse Theory (Annals of Mathematic Studies AM-51)" or Wolf's "Spaces of Constant Curvature (Ams Chelsea Publishing)" to cite but a few monumental works. Of course, if you only wanna go wandering, after learning Boothby's book, you can go safely in any direction on differential geometry, or even classical mechanics (i. e.: Synge and Schild's Tensor Calculus or Arnold's Mathematical Methods of Classical Mechanics (Graduate Texts in Mathematics)). Good luck.

This book is a standard reference on the subject of differential manifolds and Riemannian geometry in many somewhat more applied fields, such as mine (control theory). Having used it as a reference for many years, I finally decided to read it cover to cover. I'm not done yet but went through more than half. The process of reading the book in a continuous fashion, while certainly rewarding, has also led to significant disappointment. I often find flaws in the pace at which the book proceeds, in the sense that the author spends a lot of time on boring details but then goes over important material -- such as crucial steps in proofs -- too quickly and without providing sufficient insight. I also agree with another reviewer (who gave 1 star) that the often heavy notation doesn't pay off here. I have graduate training in pure mathematics so I'm used to reading books with heavy mathematical notation, but in this book things don't "click" for me and I constantly need to go back and look again for a definition of a symbol (which is often a difficult task). In addition, there is a somewhat large number of typos in the book, some of which are quite annoying.

On the other hand, it is fair to say that this book is probably as good as any other available book comparable in subject and scope. It is rigorous, mostly readable, and covers a lot of ground without being overwhelming. So, until a better one comes along, I will continue reading and using this book.

Great introductory differential geometry text! I used this book to help me pass my qualifying exam. Yay Boothby!

One day, accountants and soldiers may take an interest in differential geometry. If and when such a day comes to pass, this book will have a role to play. Until then, engineers, physicists and mathematicians alike have better alternatives, such as the inspiring texts, with complementary qualities, by Burke, "Applied Differential Geometry"; by do Carmo, "Riemannian Geometry", or by Spivak, "A Comprehensive Introduction to Differential Geometry".

Even more advanced books such as Lang's or Petersen's are more readable: in them the extra formalism brings the reward of more powerful results. Here the retentive attention to the trees at the expense of the forest is merely a barrier to entry for the uninitiated. This text's popularity in some areas of engineering must have played a role in the slow acceptance of Riemannian geometric methods.

Manuel Tenide