An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised (Volume 120) (Pure and Applied Mathematics, Volume 120) 2nd Edition by William M. Boothby (PDF)

Description

The second edition of An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised has sold over 6,000 copies since publication in 1986 and this revision will make it even more useful. This is the only book available that is approachable by “beginners” in this subject. It has become an essential introduction to the subject for mathematics students, engineers, physicists, and economists who need to learn how to apply these vital methods. It is also the only book that thoroughly reviews certain areas of advanced calculus that are necessary to understand the subject.

User’s Reviews

Editorial Reviews: From the Back Cover Differentiable manifolds and the differential and integral calculus of their associated structures, such as vectors, tensors, and differential forms are of great importance in many areas of mathematics and its applications. Although basically and extension of advanced, or multivariable calculus, the leap from Euclidean space to manifolds can often be difficult. It takes time and patience, and it is easy to become mirred in abstraction and generalization.In this text the author draws on his extensive experience in teaching this subject to minimize these difficulties. The pace is relatively liesurely, inessential abstraction and generality are avoided, the essential ideas used from the prerequisite subjects are reviewed, and there is an abundance of accessible and carefully developed examples to illuminate new concepts and to motivate the reader by illustrating their power. There are more than 400 exercises for the reader.This book has been in constant, successful use for more than 25 years and has helped several generations of students as well as working mathemeticians, physicists and engineers to gain a good working knowledge of manifolds and to appreciate their importance, beauty and extensive applications. About the Author William Boothby received his Ph.D. at the University of Michigan and was a professor of mathematics for over 40 years. In addition to teaching at Washington University, he taught courses in subjects related to this text at the University of Cordoba (Argentina), the University of Strasbourg (France), and the University of Perugia (Italy).

Reviews from Amazon users which were colected at the time this book was published on the website:

⭐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.

⭐so so

⭐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 manifolds of constant curvature and Schur’s theorem. 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 ”

⭐” or Helgason’s ”

⭐” or Kobayashi and Nomizu’s ”

⭐”, Sakai’s ”

⭐” Milnor’s ”

⭐” or Wolf’s ”

⭐” 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

⭐or Arnold’s

⭐). Good luck.

⭐When I was a doctoral student, I studied geometry and topology. At the time, I learnt about differentiable manifolds and Riemannian geometry not as a knowledge necessity, but as a background. My specialty was group theory.Groups and spaces are intimately related. In the sense, studying group theory means studying geometry (an area in mathematics studying properties of spaces). Another related area to group theory is knot theory. Although knot theory is not my specialty, I have been interested in knot theory because group theory is a useful tool in studying knots. In recent years, it has turned out that knot theory is unexpectedly related to quantum field theory in physics. That is the point I want to look into in detail in the future. To understand quantum field theory, one should have the knowledge of high-level of classical mechanics, electrodynamics, quantum mechanics, and relativity.When I started to study general relativity, I felt like to study differentiable manifold theory again. Most of all, I wanted to throughly understand more the meaning of covariant derivative. What is the meaning of a curved space? What is the meaning of differentiation in a differentiable manifold? If a covariant derivative defines a differentiation on a general manifold and we can think of a curvature at a point on it, what is the corresponding covariant derivative and curvature of surfaces in R^3? For that, I reread the differential geometry book by do Carmo and the book on Riemannian geometry by the same author, and I am really satisfied with the two books. I got definite answers. But I was not so satisfied with its logical rigor.At the time, I had several manifold theory books.1. An Introduction to Differentiable Manifolds and Riemannian Geometry, Boothby2. A Comprehensive Introduction to Differential Geometry, Spivak3. Foundations of Differentiable Manifolds and Lie Groups, WarnerAmong the three, I chose Boothby. To me, it seemed that the book is the easiest and the most reader-friendly, particularly for self-study. Moreover, I found that if one is a person with a mathematical spirit and want to study differentiable manifolds keeping studying relativity in mind, the book will be even better for him.The author’s style is philosophical, fundamental, conceptual, rather than emphasizing skills and computations. This is another merit of the book for me. For example, I solved no exercises of the book, but I had no difficulty in reading the book owing to that. But I recommend that if you ever encounter differential manifold theory for the first time, then you solve a few exercises of the earlier part of the book.For a successful reading, it is important that a reader of the book has the ability to discern what he needs and what is inessential for him now. For that, there are some prerequisites: one year course of Calculus, one semester course of linear algebra, multi-variable calculus dealing with the inverse function theorem and the implicit function theorem, and differential geometry. For the differential geometry, I recommend do Carmo. For example, if you read Boothby to know what a covariant derivative is, then you can skip the whole part about integration on a manifold.Here are detailed points.1. In the first some early sections, the author gives his effort to explain the difference between R^n and n-dimensional Euclidean space. The sections are interesting but somewhat confusing since there was no definition of n-dimensional Euclidean space. We cannot compare A with B if we don’t know what B is.2. To understand differentiable manifolds, one must know what a tangent vector is. Within the first 40 pages, the book presents three equivalent definitions of tangent vectors. They are important, so readers should carefully read the part. I don’t mean that they should follow every detail of proofs of theorems, but I mean that they should follow what the author is trying to say.3. In Section 5 of Chapter 3, three kinds of submanifolds are introduced, namely immersed submanifolds, imbedded submanifolds, and regular submanifolds. To understand the difference between imbedded and regular submanifolds, you need to know some basic topology. And I think that the arguments could be a little messy to readers. I’d like to recommend that if the arguments require too much of your time, then you take it lightly.4. Spivak’s book, Calculus on Manifolds, is a famous book about calculus on manifolds. Its level of difficulty is almost the same as Boothby’s book. But Spivak’s book is very concise (about 130 pages), so you might have already read Spivak before reading Boothby. In my opinion, in many places, Boothby is far good at introducing concepts with motivation and at clarity in its presentation than Spivak. So I recommend that you read again the corresponding part of Boothby even though you experienced Spivak’s book.5. In Section 2 of Chapter 7, there is an argument on the difference between covariant derivative and Lie derivative. In my opinion, the explanation of the section is not much helpful to understand the difference. One obvious difference is that while the Lie derivative of two vector fields is defined on any differentiable manifold, a covariant derivative of two vector fields needs an additional structure on a differentiable manifold. Although there is an explicit and computational relationship between them, I don’t think that that’s all. I guess that it could be a theme of a paper.6. If you have read do Carmo’s Riemannian geometry and thought that the proofs on covariant derivatives are sloppy since the author uses local concepts in proving global ones, then Boothby’s book will be the best guide for you. The key is to use the partition of unity.It took me about four weeks to read almost the whole book without studying anything else. Considering that I skipped many sections (for example, sections on Frobenius’s theorem, too detailed theorems on Lie groups, and de Rham groups. And covering space theory and fundamental groups that I already know) and I was already familiar with differentiable manifold theory, I think that I was not so speedy. I hope that I will be faster next time.I love the book, but it is not perfect. There are some typos and in a few places there seems to be a little messy arguments. But the book has overwhelmingly more good points. I really enjoyed the book, and it was beneficial. I appreciate the author.

⭐Boothby’s book is now a classic. It serves best for an absolutely reliable reference book of an undergraduate course in Differential Geometry of manifolds. For a graduate course, it is rather insufficient; newer books can play this role much better but still, one has to have Boothby next to them.The older edition had numerous typos; this has been corrected substantially.

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