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Customer Reviews:
Very Nice Nontrivial Introduction
By Kevin R. Vixie - May 31, 2000This 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.
This is a book for REAL mathematicians
By T. Baillie Arnold - April 16, 2005This 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.
OK but not great
By Daniel - September 9, 2011This 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... read more
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