Classification of plane curves under the group of special affine motions. Classification of curves in Euclidean n-space.
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Meusnier's Theorem. Gaussian curvature. The Theorema Egregium. The metric in geodesic polar coordinates. The formula of Bertrand and Puiseux; Diquet's formula.
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The form of the metric in Riemannian normal coordinates. Sectional curvature. The Test Case; first version. Finsler metrics. Ricci's Lemma. Ricci's identities. The curvature tensor. The Test Case; second version. Classical connections.
The torsion tensor. Bianchi's identities.
Covariant derivatives. Parallel translation. The Levi-Civita connection.
The Test Case; third version. The First Variation Formula. Connections with the same geodesics. The structural equations of Euclidean space. The structural equations of a Riemannian manifold. Adapted frames. The Test Case; fifth version. The Test Case; sixth version. The 2-dimensional case. Cartan connections. Conformally equivalent manifolds. Cartan's treatment of normal coordinates.
Lie groups acting on manifolds. Ehresmann connections. The dual form and the torsion form. The structural equations. The torsion and curvature tensors. The Test Case; seventh version. 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.enter
Mathematics Textbooks for Self Study -- A Guide for the Autodidact
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.