[Math] Prerequisites for studying smooth manifold theory

Question

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)

Answer 1

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.