Voodoonris Chris Shaver rated it liked it May 20, Just a moment while we sign you in to your Goodreads account. It is also the only book that thoroughly reviews certain areas of advanced calculus that are necessary to understand the subject. The second edition of An Introduction to Differentiable Manifolds and Riemannian Geometry, Revised has sold over 6, copies since publication booyhby and this revision will make it even more useful. Account Options Sign in. Thus a smooth surface, the topic of the B3 course, is an example of a 2-dimensional manifold. Partitions of unity, integration on oriented manifolds.

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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. 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 —7: the ratio of elementary to advanced material is too large. Spivak, M. Publish or Perish, 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 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. First edition: Birkhauser, The first edition was the textbook for MTG in I have not looked at the more recent edition. Lang, S. Springer-Verlag, 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.

The library has the version and one or more of the earlier editions, as well as the book. Loomis, L. Addison-Wesley, This text was used in my first introduction to manifolds as a student. Nicolaescu, L. World Scientific, Very nice presentation and progression of topics from elementary to advanced.

Benjamin, NY, Sternberg, S. Lectures on Differential Geometry. Warner, F. Scott Foresman ; reprinted by Springer 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, ; reprinted later by Dover. Noel Hicks, Notes on Differential Geometry. Van Nostrand, Nice, short 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.

Prentice-Hall, Hirsch, M. Milnor, J. University Press of Virginia, later editions published through at least A 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. 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. A beautiful book but presumes familiarity with manifolds. Cheeger, J. Kobayashi, S. A classic reference, considered the bible of differential geometry by some, especially for the material on connections in vol.

The first page of vol. Part III is an excellent treatment of the geometry of geodesics. Academic Press, Geometry in the presence of a general indefinite or definite metric. Last update made by D.

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## An introduction to differentiable manifolds and Riemannian geometry, Volume 63

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