Fundamentals of Differential Geometry (Graduate Texts in Mathematics, 191)

This is 'Lang' - what else can be said! However if you see the need to work on infinite dimensional tangent bundles (which I do) the author's very broad setting allows you to see those details which are structural as separated from details which are technical and specific to context beyond the general theory. From this point of view always useful.

There are many pragmatic books on differential geometry which have quite definite practical purposes, such as applications to physics (cosmology and particle physics), to the Poincaré conjecture (now a theorem) and related theorems relating geometry to topology, and to engineering and optics. This book, " Fundamentals of Differential Geometry ", by the exceptionally prolific Serge Lang, is useful as background for such practical purposes, but I would characterize its main focus as the "high art" or "high culture" of differential geometry.For example, Lang presents almost all of differential geometry with general Banach spaces for the coordinate space instead of the usual finite-dimensional Euclidean spaces. Most of the 1998 Foreword is a justification by Lang of the infinite-dimensional framework for differentiable manifolds which he assumes throughout. For readers who are unfamiliar with topological vector spaces, and Banach and Hilbert spaces in particular, Lang provides a 4-page summary of the whole subject (pages 5-9). This will seem somewhat inadequate if you have not studied this subject before.Likewise, Lang assumes a framework of category theory throughout. Lang uses category theory language in many situations where plain mathematical words would have been much clearer. But Lang's strong background in the culture of algebra seems to have predisposed him to use category theory whenever possible. Therefore in this aspect also, the reader will need to have a fairly good background in basic category theory before commencing this book. Lang provides a one-and-a-quarter page summary of category theory (pages 4-5), which will be inadequate for beginners.Whenever I have dipped into this book to find an alternative explanation for something which I was having difficulty with in the other 45 differential geometry books on my bookshelf, I have found it takes up to a week to work out what Lang is saying. All of the definitions have to be traced back through other definitions, which have to be traced back, and back, and back, until they can be translated into the standard definitions in the pragmatic DG literature. For example, this week I am looking for proofs of the existence and uniqueness of parallel transport for a given affine connection, and I want the constraints on the connection to be as weak as possible. I'd like the theorem to work for general rectifiable or integrable connections. So I look up Lang's theorems for this, but to understand them, I need to trace his rather unusual definitions from page 206 back through the previous 200 pages to find out how he defines connections, parallelism, time-varying vector fields (on Banach spaces), and the mean value theorem. It's a bit like asking someone what time it is and they insist on giving you a three-hour summary of the history of chronometry and the multiple definitions of atomic time, local time, universal time, ephemeris time and siderial time before finally looking at their watch and telling you what time it is in ancient Sumerian. Since I am that kind of person too, I guess I can't complain. But the acquirer of this book by Serge Lang should be aware that this is not an introduction to pragmatic differential geometry. It is high-culture mathematics for the sake of furthering the high art of mathematics. Despite this, it does contain many useful theorems, definitions and explanations which are of practical utility.This book seems to be a superset of all of the other books by Serge Lang on differential geometry. The following is what I have been able to ascertain.* Chapters I to IX, and XV to XVIII, are the same as in Lang's 1995 " Differential and Riemannian Manifolds ". (The copyright page of this 1995 publication says that it is the 3rd edition of Lang's 1962 book, " Differential Manifolds ".) So the 1995 book is missing chapters X to XIV.* Chapters I to VII, and XVI to XVIII, are in Lang's 2002 " Introduction to Differentiable Manifolds ". So the 2002 book is missing chapters VIII to XV.Conclusion:This is not a pragmatic book. It is not a beginner's book. It is a book to be read as preparation for doing research in differential geometry of a fairly abstract kind. It is also useful as a reference when doing research.* PS. 2016-5-16.Since I first wrote the above review in July 2014, I have found this book more and more useful. The more I read it, the less mysterious it becomes. And finally it is no longer "high culture" to me, but rather a very practical book. I think that some readers might be put off by the continual use of Banach spaces where 95% of authors would have finite-dimensional Cartesian spaces, the rest is really a very wide-ranging coverage of modern differential geometry.

See my review of the  hardcover version of this book , which I bought in May 2007 directly from Springer. Even though I paid the full price for it, I regard this book as one of my wisest purchases amongst the 45 DG books in my personal collection. (These are all real DG books, not "e-books".)

This book looks more like a big collection of essays than really a treatise on one subject. If you have read other books by Lang you will notice that some material included in previous works appears reproduced here word by word. Also, there are some inconsistencies in the notation of different sections, making it obvious that different parts of the book were written at different times, and perhaps by different persons.All these shortcomings don't mean that the book is bad. Quite the opposite: It is a very complete survey on modern differential geometry, including from the fundamentals up to recent results. The graduate student and the working mathematician will find it very useful.