I've included comments on some of the books I know best; this does not imply that they are better than the other books on this list. (Nor should one conclude anything from the order in which the books are listed—alphabetical by order within each group—or by comparing the lengths of different comments.)
General references that do not require too much background
- Boothby, W., An Introduction to Differentiable Manifolds and Riemannian Geometry, 2nd ed., Academic Press, 1986.
- Advantages: This book gives a thorough treatment of the most basic concepts of manifold theory, and a good review of the relevant prerequisites from advanced calculus. Many examples are given. Disadvantage as a textbook for MTG 6256–7: the ratio of elementary to advanced material is too large.
Spivak, M., A Comprehensive Introduction to Differential Geometry, volumes 1 and 2. Publish or Perish, 1979 .- Advantages: Spivak's encyclopedic five-volume series ("Big Spivak") is very down-to-earth. More motivation and historical development is given here than in any other text I know. Disadvantage as a textbook for any course: Too much detail; volume 1 alone is 668 pages. One learns better if more is left to the reader. Also, the fact that these books were photocopied from the typewritten manuscript (rather than typeset) can make for difficult reading.
Conlon, L., Differentiable Manifolds: A First Course. First edition: Birkhauser, 1993. Second(?) edition: 2008.- The first edition was the textbook for MTG 6256 in 1997. It is similar in many ways to Boothby's book above, but is written at a more sophisticated level, and covers some topics omitted by Boothby. I have not looked at the more recent edition.
Lang, S., Differential and Riemannian manifolds. Springer-Verlag, 1995.- This is an updated version of Lang's older book "Differential Manifolds", which is one of the most commonly cited references for fundamentals in this area. The treatment is elegant and efficient. However, Lang writes in the generality needed for infinite-dimensional manifolds, requiring some comfort with infinite-dimensional Banach and Hilbert spaces on the part of the reader. For a first course in manifolds, this may be daunting and may hinder the development of intuition. In our class, we will stick to finite-dimensional manifolds, at least in the fall semester, and probably in the spring as well.
Lang also has a 1999 book called "Fundamentals of Differential Geometry", which despite the different title seems to be just the most recent version of "Differential Manifolds". The library has the 1995 version and one or more of the earlier editions, as well as the 1999 book. The library also has a link to an e-book entitled "Introduction to differentiable manifolds", which I have not looked at yet but probably has significant overlap with Lang's other books above.
Loomis, L., and Sternberg, S., Advanced Calculus. Addison-Wesley, 1968.- Chapters 9, 10, 7, and 11 (11 requires 7) cover the basics of manifolds up through Stokes' Theorem. This text was used in my first introduction to manifolds as a student. I didn't understand anything about manifolds until three days before the final exam, but once it sunk in I found this book an excellent reference.
Nicolaescu, L., Lectures on the Geometry of Manifolds. World Scientific, 1996.- Very nice presentation and progression of topics from elementary to advanced.
Spivak, M., Calculus on Manifolds. W. A. Benjamin, NY, 1965.- "Little Spivak" is a beautiful, very self-contained 146-page treatment of advanced calculus on Rn and on manifolds. It starts with the definition of the derivative on Rn and gets all the way through Stokes' Theorem on manifolds.
Sternberg, S. Lectures on Differential Geometry. Prentice Hall 1964; 2nd edition by Chelsea 1983; reprinted later by AMS.
Warner, F., Foundations of Differentiable Manifolds and Lie groups. Scott Foresman 1971; reprinted by Springer 1983 in hardcover, and again later in softcover.Books in the next group go only briefly through manifold basics, getting to Riemannian geometry very quickly.
- Richard L. Bishop and Richard J. Crittenden, Geometry of Manifolds. Academic Press, 1964; reprinted later by Dover.
Noel Hicks, Notes on Differential Geometry. Van Nostrand, 1965.- Nice, short (183 small pages), and out of print.
Books in the next group focus on differential topology, doing little or no geometry.
(Remember that differential geometry takes place on differentiable manifolds, which are differential-topological objects. Some of the deepest theorems in differential geometry relate geometry to topology, so ideally one should learn both.)
Guillemin, V, and Pollack, A., Differential Topology. Prentice-Hall, 1974.
Hirsch, M., Differential topology. Springer-Verlag, 1976.
Milnor, J., Topology from the Differentiable Viewpoint. University Press of Virginia, 1965 (later editions published through at least 1976).- A 64-page gem.
Less elementary books.
These either assume the reader is already familiar with manifolds, or start with the definition of a manifold but go through the basics too fast to be effective as an introductory text.
Beem, J., Ehrlich P., and Easley, K., Global Lorentzian Geometry. 2nd edition, Marcel Dekker, 1996.- An excellent reference for the mathematics of general relativity: geometry in the presence of a Lorentz metric (indefinite) as opposed to a Riemannian metric (positive definite).
Bott, R., and Tu, L., Differential Forms in Algebraic Topology. Springer-Verlag, 1982.- A beautiful book but presumes familiarity with manifolds.
Cheeger, J., and Ebin, D., Comparison Theorems in Riemannian Geometry North-Holland, Amsterdam, 1975.
Kobayashi, S. and Nomizu, K., Foundations of Differential Geometry vols. 1 and 2, John Wiley & Sons 1963.- A classic reference, considered the bible of differential geometry by some, especially for the material on connections in vol. 1. The first page of vol. 1 scares away many a reader, but from page two on the book is quite readable.
Milnor, J., Morse Theory. Princeton University Press, 1963 (Annals of Mathematics Studies v. 51).- A short and very readable classic, of which parts II and III are relevant to this course. The 24-page Part II, "A rapid course in Riemannian Geometry," is accurately named. Part III is an excellent treatment of the geometry of geodesics.
O'Neill, B.,Semi-Riemannian Geometry: with Applications to Relativity. Academic Press, 1983.- Geometry in the presence of a general indefinite or definite metric.