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I have studied differential geometry, and am looking for basic introductory texts on Riemmanian geometry. My target is eventually Kähler geometry, but certain topics like geodesics, curvature, connections and transport belong more firmly in Riemmanian geometry.

I am aware of earlier questions that ask for basic texts on differential geometry (or topology). However, these questions address mainly differential geometry. I'm more interested in Riemmanian geometry here.

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If you want to devote to geometry, I strongly recommend the Lectures in Geometry by Postnikov, rather deep yet clear. His book consist of 6 volumes available in French and Russian.(It's a pity that we do not have volume 4 in English) And what horrified me was the fact that this series of books had been given to Russian undergraduates. –  Henr.L Aug 1 '13 at 12:31
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10 Answers

up vote 19 down vote accepted

Personally, for the basics, I can't recommend John M. Lee's "Riemannian Manifolds: An Introduction to Curvature" highly enough. If you already know a lot though, then it might be too basic, because it is a genuine 'introduction' (as opposed to some textbooks which just seem to almost randomly put the word on the cover).

However, right from the first line: "If you've just completed an introductory course on differential geometry, you might be wondering where the geometry went", I was hooked. It introduces geodesics and curvature beautifully and is very readable.

I think the first chapter might be available on the author's website.

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The entire trilogy by Lee-INTRODUCTION TO TOPOLOGICAL MANIFOLDS,INTRODUCTION TO SMOOTH MANIFOLDS and this one-are worth having,Spenser.Not only are they beautifully written by a true expert and terrific teacher,they are probably the friendlest texts any student will ever find at this level. YES,they ARE too verbose at times,especially the second volume-but for thier sheer beauty and care for the reader's sensibilities,they can be forgiven.The introduction to "Smooth Manifolds" says "To my students". It's clear with all 3 books,Lee is saying "to my students WITH LOVE." –  Andrew L Mar 27 '10 at 21:36
    
I agree that this is a great book. But beware that the first edition (the only one available right now, although apparently a second one is in progress) has a fair number of errors. Fortunately you can find the errata available here: math.washington.edu/~lee/Books/riemannian.html at Lee's website. –  MTS Feb 8 '13 at 22:32
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One more vote for Gallot, Hulin, Lafontaine. I think this book does a better job of most of presenting clean proofs (including avoiding the use of co-ordinates and Christoffel symbols) and a more geometric approach than other books, which tend to get bogged down in the abstract formal computations. A lot of important explicit examples are worked out in detail. It also shows very nicely how curvature bounds can be used with Sturm-Liouville theory applied to Jacobi fields along a geodesic to establish global geometric properties of a Riemannian manifold. This is the heart of global Riemannian geometry as developed by Berger, Toponogov, and others and raised to a high art by Gromov and Perelman among others. But you wouldn't know that from many other books on Riemannian geometry.

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I like do Carmo's Riemannian geometry.

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So do I,Jose. Very nice rapid into to manifolds.The student pressed for time will like this one very much. –  Andrew L Mar 27 '10 at 21:37
    
So do I. Clear and rigorous. –  Joël Dec 16 '10 at 16:29
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I do to, but I've had some students complain that it doesn't have enough detail. –  Jim Conant Dec 3 '11 at 20:00
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I'd like to add O'Neil's Semi-Riemannian Geometry, with applications to relativity. The "semi" stuff is safely ignorable if you only want Riemannian Geometry (i.e. you can always simply ignore the prefix "semi-" and specialise to positive definite), and if you have a mildly physicsy leaning, it's nice to have the relativistic connections laid out.

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THIS IS A FANTASTIC BOOK AND IT SHOULD BE IN EVERY MATHEMATICIANS' LIBRARY.IT SHOWS HOW BENIFICAL A UNIFIED APPROACH TO BOTH MATHEMATICS AND PHYSICS CAN BE. IT'S A CRIME IT'S OUT OF PRINT. –  Andrew L Mar 27 '10 at 21:32
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There is a chapter in Milnor's Morse Theory that covers the basics.

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I like do Carmo's Riemannian geometry, which is more down-to-earth, and gives more intuition.

Jurgen Jost's Riemannian geometry and geometric analysis is also a good book, which covers many topics including Kahler metric.

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Of course, the book of Gallot et al is very nice. But if you are intresseted in Kaehler manifolds you should also look into Besse: Einstein manifolds.

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I would differenciate between some books. If you simply want to learn how to calculate things, for instance because you are interested in physics, I'd recommend to you the book "Geometry, Topology and Physics" by Nakahara which has a very good part on Riemannina goemetry. However, it may contain more than you actually want to know. The book with which I have really understood Riemannian Geometry was Jürgen Jost's "Riemmanian Geometry and Geometric Analysis". However, it might require some prerequisites that you haven't got so far. Alternatively, you could read Do Carmo's book. It is very concisely written and features some nice examples. I would not advise you to read the Gallot-Hutin because some of the proofs are simply only outlined. For a person that begins learning Riemannian geometry this could be very discouraging. However, I admit that the great advantage of this book is the number of exercises.

Something I want to add because I really want to advertize one book: Riemmanian geometry by Takahashi sakai. If you once have got the basics, this books takes you further. A quantum leap further. But you really need the basics. I would use this book for a second course in Riemmanian Geometry, assuming the student's familiarity with differentiable manfiolds and fiber bundles and a first course in Riemannina Geometry, such as for instance material covered in Jost's book in the chapters 1-4.

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I had great trouble finding a single book that would be good for everything. There is a book Lectures on Differential Geometry by Chern, Chen, and Lam that's pretty nice (although Chern's name on the cover might be affecting my judgment). It has the advantage of being very concise and rather clear.

EDIT: The question asked specifically for Riemannian geometry rather than differential geometry. If I were to describe the above book, I'd say it's mostly about the former, regardless of the title (although it has a few chapters on other topics at the end). However, I'm not sure I understand the difference well enough to judge. It certainly has a chapter on "Riemannian geometry".

(Also, I second the suggestion of Milnor's Morse Theory. The appendix to Milnor's Characteristic classes has a very nice exposition of connections, but it has no other Riemannian geometry).

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Well,ANY version of Chern's notes are well worth having,Ilya-but they are very concise,advanced (you really need a year of topology and algebra at the graduate level to fully understand them) and they have the HUGE down side of no exercises.If someone could supply exercise sets-preferably one of Chern's former students-the book would be much more useable as a text.To be honest,I think any reader that tries to learn DG from them without a good course in classical curve/surface theory is going to be VERY lost. –  Andrew L Mar 27 '10 at 21:39
    
@Andrew: I agree that the lack of exercises is a downside. Regarding the level, it certainly uses a lot of differential forms. This seemed quite enlightening to me (concerning the forms more than about the geometry), but if one fears them strongly enough (as I certainly have at some point) then it could be a problem. –  Ilya Grigoriev Mar 27 '10 at 22:28
    
Yes,it does use a lot of differential forms and without a considerable background in algebra (at the very least,a strong algebra course at the undergraduate level a la Herstien or Artin),this is going to be a big hurdle to overcome.A much gentler and geometric introduction to forms-as well as a terrific first course on DG-is provided by the second edition of Barrett O'Niel's ELEMENTARY DIFFERENTIAL GEOMETRY. Indeed-a year long course on curve and surface theory based on O'Niell would make Chern's notes MUCH easier to absorb. –  Andrew L Mar 28 '10 at 1:24
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You may look into Novikov's 3 volumes of differential geometry

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