A professor recently told me that if I can cover the chapters on curvature in Petersen's Riemannian geometry book (linked here) within the next few months then I can work on something with him. However, before I plan on reading this book I need to pick up some manifold theory. I plan on reading Lee's Smooth Manifolds book.

Assuming I don't know anything covered in Lee's book and that I want to read it just to be able to read Petersen's can somebody recommend me what chapters I can skip. I know it'll be beneficial to read the whole book but I'll cover the whole book in a course next year and I just want to be able to read the rudiments of Riemannian geometry before then. Or if one can suggest a book better suited for this goal that would be appreciated too. Thank you.

  • $\begingroup$ Instead of Lee, you could read Helgason's Symmetric Spaces book, the first 100 pages cover basic differential geometry. $\endgroup$ – Carsten Apr 23 '14 at 14:20
  • $\begingroup$ Milnor's "Topology from the Differentiable Viewpoint" is worth reading for a quick introduction to smooth manifolds that gets rapidly to some substantial theorems. $\endgroup$ – Dan Fox Apr 23 '14 at 15:30

Petersen's book is challenging, but very clear and thorough. If you want to learn the prerequisites quickly-as I'm sure all graduate students who want to begin research do-then John Lee's books aren't really the best option for you. They're wonderfully lucid and comprehensive texts,but thier sheer length means they're really best for self study when you have several months to invest. If you need a faster introduction, Loring Tu's An Introduction To Manifolds is a better choice. Unfortunately it avoids basic topology, so you're going to need to supplement it with a topology text. John McCleary's A First Course In Topology is short, beautifully written and probably has everything you need. Lastly, I'd be remiss if I didn't call to your attention Petersen's own notes on basic manifold theory, which he's made available at his website: http://www.math.ucla.edu/~petersen/manifolds.pdf They're more sophisticated then Tu and they'll make wonderful collateral reading.

That should get you started and ready for your advisor by summer. Good luck!

  • $\begingroup$ What do you mean by "avoids topology"? General topology is required for anything beyond the first chapter of Loring Tu's book. Furthermore, the book itself contains a quick review of some general topology topics in the appendix. $\endgroup$ – Ayman Hourieh Apr 23 '14 at 17:31
  • $\begingroup$ Dear @Ayman, I think Mathemagician1234 means that Tu's textbook assumes general topology as a prerequisite, which is why Mathemagician1234 has typed "so you're going to need to supplement it with a topology text" in the following sentence, and proceeds to give a specific example of such a topology text. $\endgroup$ – Amitesh Datta Apr 23 '14 at 23:46
  • $\begingroup$ +1 for several great suggestions. I have never heard of Petersen's own notes (the ones available on his website), and they are definitely different in terms of coverage compared to most introductions to smooth manifold theory. However, I think Chapters 1 and 2 (and maybe 3) look really good if one's goal is to get to Riemannian geometry as quickly as possible. Certainly, chapter 4 also contains some important algebraic topology which would be useful to any one specializing broadly in topology/geometry. $\endgroup$ – Amitesh Datta Apr 23 '14 at 23:52
  • $\begingroup$ @Ayman@Amitesh:First,thanks for the nice comments,Amitesh. To clarify-One of the goals of Tu's book is to provide a broad introduction to differentiable manifolds with very minimal prerequisites. He bemoans in the introduction "the heavy burden of point set topology in most manifolds texts". So the only explicit prerequisites are undergraduate abstract algebra and real analysis. I really like Tu's book, but this delibrate attempt to avoid topology really hurts his presentation sometimes-particularly towards the end and the discussion of the De Rham cohomology without the fundamental group. $\endgroup$ – Mathemagician1234 Apr 24 '14 at 17:16
  • $\begingroup$ @Mathemagician1234 Thanks for the clarification. I disagree that Tu attempts to avoid general topology. For example, chapter two ends with quotient spaces, which is heavy on topology. I also disagree that one needs to introduce the fundamental group before de Rham cohomology. The fundamental group can be a nice intro to homotopy groups or singular homology, but it's not really needed for differential forms, which are used in de Rham cohomology. $\endgroup$ – Ayman Hourieh Apr 24 '14 at 19:42

I really liked "Riemannian Geometry" by Manfredo do Carmo - http://www.amazon.com/dp/0817634908/ref=rdr_ext_sb_ti_sims_1 (a PDF copy can be found by googling "Riemannian geometry do Carmo" and looking at the first few search results)

Chapter 0 discusses the preliminaries from smooth manifold theory and subsequent chapters immediately begin in Riemannian geometry (Chapter 0 is only roughly 30 pages in length and the entire book is roughly 300 pages in length). Of course, one should be warned that Chapter 0 is quite terse and I think it is better to have some familiarity with smooth manifolds beforehand. However, with enough mathematical maturity, it should be possible to learn Riemannian geometry from do Carmo without any background in smooth manifold theory, beginning with Chapter 0.

Also, another alternative is "An Introduction to Differentiable Manifolds and Riemannian Geometry" by William Boothby - http://www.amazon.com/Introduction-Differentiable-Manifolds-Riemannian-Mathematics/dp/0121160513

I really liked this book - it covers both smooth manifold theory (at roughly the level of Lee but in the space of 300, rather than 500 pages) and also covers Riemannian geometry in two chapters. The depth of coverage in Riemannian geometry is not very much but the coverage of smooth manifold theory is quite efficient compared to Lee. You could just read chapters 1 - 5 (roughly 200 pages with relatively large font) and skip chapter 6 (chapters 7 and 8 are on introductory Riemannian geometry, which you can read if you like, or move onto a more specialised textbook in the subject such as Peterson or do Carmo).

Take a look and let me know what you think!

  • $\begingroup$ Lee's "Riemannian manifolds" book (different from his "Smooth manifolds" book) is similar in spirit, level, and length to do Carmo's. $\endgroup$ – Dan Fox Apr 23 '14 at 15:24

First a disclaimer: I have never read Petersen's book, so I'll answer based on having skimmed through the contents and first chapters and assuming it is not much different from other books of the same level. Therefore take this with a grain of salt.

I really like Lee's Smooth Manifolds, but since you want a brief introduction so you can study Riemmanian geometry, and you will take a course in manifolds, I would strongly suggest considering another book. Since Lee's is very complete you would have to read the first 16 chapters (possibly omitting cp. 7 and 13) so you cover basic structures, bundles and differential forms. That's over 300 hundred pages.

I'll second Amitesh Datta on Boothby's book, which is great. If you want an even shorter introduction I think is hard to do better than "Geometry of Manifolds" by Richard Bishop and Richard Crittenden: http://www.amazon.com/Geometry-Manifolds-AMS-Chelsea-Publishing/dp/0821829238

The first 6 chapters (~100 pages) gets you everything you need to start doing Riemmanian geometry, and if you like the book you can go on with it too. Although I don't recommend as the place to start, when I first studied manifolds I found very useful to have nearby Kobayashi and Nomizu's "Foundations of Differential geometry vol. 1": http://www.amazon.com/Foundations-Differential-Geometry-Classics-Library/dp/0471157333. The first 4 chapters (~150 pages) have what you need, so maybe you'll give it a shot when something is eluding you.

  • $\begingroup$ +1 for several great suggestions. I especially think Bishop and Crittenden looks like an excellent textbook in terms of the combination of its coverage and terseness. $\endgroup$ – Amitesh Datta Apr 23 '14 at 23:49

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