Noud's Reviews > Introduction to Smooth Manifolds

Introduction to Smooth Manifolds by John M. Lee
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U_50x66
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Aug 18, 12

Read from July 22 to August 18, 2012

Introduction to Smooth Manifolds from John Lee is one of the best introduction books I ever read. I read most of this book, except for the appendices at the end and proofs of some corollaries. This book covers a couple of subjects:
(*) First the theory of smooth manifolds in general (ch1, 2, 3, 4, 5 and 6), smooth maps, (co)tangent spaces, (co)vector fields and vector bundles. These first few chapters contain a lot of examples. These six chapters can be used as a first introduction course to smooth manifolds. Basically only tools from topology, linear algebra and calculus are used here.
(*) The second major part is about sub-manifolds (ch7, 8, 9 and 10), theorems like the Inverse Function, Implicit Function and Rank theorem are covered. Here the definition of Lie groups is introduced. Also Whitney's approximation theorems are discussed.
(*) The third major part is about integration on manifolds (ch11, 12, 13 and 14). These tools are constructed from scratch. There is a lot of attention to Tensors, Riemannian manifolds, differential forms and orientation on manifolds. In ch14 Stokes theorem is proved, a nice milestone in Differential Geometry.
(*) Chapters 15 and 16 can be viewed as a appendix of the third major part. In these sections the Rham theory is developed, homology and cohomology with the Rham Theorem as main result.
(*) The last part (ch17, 18, 19, 20) explores the circle of ideas surrounding integral curves and flows of vector fields. Also a lot of new theory for Lie groups and Lie algebras is introduced and developed in this section.

This book is very well written. It contains a lot of information (it's 628 pages long!) but is very well organized. Every new concept starts with a good motivation, mostly based on the "ordinary" case in R^2, R^3 or R^n and is then developed in more generality. In this way it is good didactic book and in my opinion suitable for one or more courses on Differential Geometry. Since everything is well organized and it contains a lot of information it can also be used as a reference. I encourage everyone who has some interests in differential geometry to buy this book and at least read some sections of it. 5/5 stars!
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Reading Progress

08/14/2012 page 274
42.0% "Damn hot stuff is happening here..."

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