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I am attending first year graduate school in about three weeks and one of the courses I am taking is an introduction to smooth manifolds. Unfortunately, my topology knowledge is minimal, limited to self study. Besides some basic topological definitions, are there specific areas where I should become acquainted with? I have heard multivariable calculus is a 'prerequisite' for the study of smooth manifolds -- is this from an intuition perspective, and if so, what parts and to what degree of rigor are they referring?

Thank you. (My apologies if this is a repeat question, by the way -- I could not find it)

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You need multivariable calculus, because you need to understand what it means for a function $\mathbb{R}^m\to\mathbb{R}^n$ to be differentiable. –  M Turgeon Aug 6 '12 at 15:38
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Open up a book on topological manifolds and see what topology they use. –  Qiaochu Yuan Aug 6 '12 at 15:39
    
Well, I have and so I understand I need to understand several topological concepts: Hausdorff, bases, countability, etc., but I suppose I am wondering if there is anything else I need, perhaps only to help my intuition while learning, or more. –  Three Aug 6 '12 at 15:43

3 Answers 3

up vote 5 down vote accepted

You will need

  • Calculus/real analysis of functions of one and several variables up to and including the implicit and inverse function theorem. This (the implicit function theorem) is the basic starting point for (smooth) manifold theory, you will not be able to get anywhere without it.
  • sound knowledge of linear algebra (at least in/for real vector spaces)
  • 1-d integration (Riemann integral will suffice in the beginning), preferably basic knowledge of an n-dimensional integration theory.
  • basic theory of ODE will sooner or later become important, in order to be able to deal with, e.g, the flow of vector fields (actually one needs existence (Picard-Lindelöf) and rather soon smooth dependence of the solution on the initial values, but the latter may be stated and believed. It'll become difficult without existence of solutions).

Depending on your book of choice everything else will probably be developed in the course of said book, see the other answers for some suggestion. The basics in topology you'll need will likely be known to you, once you really covered the above list. Spivak's calculus on manifolds comes to my mind.

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Very helpful -- thank you. –  Three Aug 6 '12 at 16:39

In my home University, the textbook used for the first graduate course on Manifold theory is Introduction to Smooth Manifolds, by John M. Lee. If you look it up at your library, you will see there is an Appendix with three sections: Topology, Linear Algebra, Calculus. This Appendix will give you some prerequisites you are looking for.

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They usually refer to using inverse function theorem, etc to construct smooth manifolds, and to my knowledge this is not included in Stewart calculus book. There are other topics usually included in a differentiable manifolds book but not covered in calculus too, like Lie derivative, homogeneous spaces, Poincare-Hopf theorem, etc.

The classical reference on smooth manifolds is Boothy's book, and you may read it online or grab a copy yourself. The other book by Lee others mentioned is also good for the purpose, and probably covered more topology than Boothy's book. I read them several years ago and do not remember the strucutre very well. You may be interested to read Munkres's Analysis on Manifolds as well.

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I am quite enjoying Boothy's book. It's a good introduction. Thank you. –  Three Aug 6 '12 at 16:40

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