I have studied differential geometry, and am looking for basic introductory texts on Riemannian 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 Riemannian geometry here.
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.
*like this* rather than shouting. :)
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Commented
Mar 27, 2016 at 17:39
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.
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.
There is a chapter in Milnor's Morse Theory that covers the basics.
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.
Of course, the book of Gallot-Hulin-Lafontaine is very nice. But if you are interested in Kaehler manifolds you should also look into Besse: Einstein manifolds.
I would differentiate 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 Riemannian 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-Hulin-Lafontaine 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 advertise 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 manifolds and fiber bundles and a first course in Riemannian Geometry, such as for instance material covered in Jost's book in the chapters 1-4.
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).
You may look into Novikov's 3 volumes of differential geometry
In the introduction to Lee's book there is a reference to "Frank Morgan’s delightful little book" (Frank Morgan. Riemannian Geometry: A Beginner’s Guide. Jones and Bartlett, Boston, 1993.) That book is, indeed, delightful.
All of Isaac Chavel's books are excellent, but in particular Riemannian Geometry: a Modern Introduction is a great book if you are already comfortable with elementary differential geometry.