What math is necessary to learn manifolds? I want to learn about manifolds, but I'm only a senior in high school and obviously have a while to go.  I'm in AP Calc BC.  What should I study to eventually learn manifolds? Linear Algebra? What else?
 A: You should most likely have a background in linear algebra as you said, multivariable calculus, and real and complex analysis.  For instance, Rudin begins a discussion of manifolds after discussing standard multivariable analysis in his Principles of Mathematical Analysis.
A: I think the prerequisites you will need to study manifold theory depend on which aspect of manifold theory that you wish to study. For example, a good knowledge of algebraic topology is more essential if you wish to study differential topology than if you wish to study differential geometry (although you should eventually learn algebraic topology in some depth no matter which aspect of manifold theory you pursue). However, the prerequisites to study the standard theory of differentiable manifolds are (generally speaking) point-set topology, linear algebra and advanced (multivariable) calculus. A good knowledge of point-set topology and linear algebra implies that you have the mathematical maturity necessary to study manifold theory as well as the necessary knowledge; thus, it is important to carefully study these two subjects. 
For example, you might wish to look at Topology: A First Course by James Munkres and Linear Algebra Done Right by Sheldon Axler which will provide you with more knowledge in these subjects than is strictly necessary in manifold theory (but this knowledge will be essential in your study of other branches of mathematics). Finally, I think Principles of Mathematical Analysis by Walter Rudin furnishes a solid knowledge of the elements of advanced calculus (both single-variable and multivariable) that will be necessary for manifold theory.
In short, it would be a good idea to use manifold theory as a means to advance your knowledge of other (essential) branches of mathematics because the prerequisites for manifold theory are more fundamental in mathematics (as a whole) than manifold theory itself. 
I hope this helps!
A: I agree with @analysisj 's response (and up-voted it).  I just wanted to add that it might be useful to review some topology depending on how your real and complex analysis courses were taught.  Some analysis courses use books that are light on the point set topology theory. 
A: In addition to linear algebra, analysis and topology as others have suggested, learning some classical differential geometry probably wouldn't be a bad idea either. The very abstract definitions one encounters in the theory of manifolds are inspired by the principles of differential geometry much as point-set topology was inspired by analysis. Seeing these concepts made tangible by concrete calculations will give more meaning to the more elaborate machinery of manifolds and differential forms.
A: Here's a nice basic introduction -- it gives an idea of what manifolds are, and you don't need to know linear algebra or multivariable calculus to read it: http://www.math.washington.edu/~lee/Books/ITM/c01.pdf
A good resource for people who haven't even had calculus but who want to learn some basic ideas in topology is http://www.learner.org/courses/mathilluminated/units/4/textbook/01.php
Before you can even learn the precise definition of manifolds and theorems about manifolds you should be familiar with topological notions that students typically learn in analysis. It's difficult to have any intuition behind manifolds if you don't have a good idea of $\mathbb{R}^n$. But you should take a look at Chapter 2 of Munkres's Topology. Try to understand what you can. If you get too lost, consult an introductory analysis book like Rosenlicht's Introduction to Real Analysis and hop back and forth between the two. If you want to do calculus on manifolds, you should certainly learn some linear algebra and multivariable calculus first.
A: I hate to add yet another answer to the list, but I would like to be thorough.  The pre-requisites for learning about manifolds are as follows:


*

*Multivariable calculus

*Linear algebra

*Real analysis

*Point-set topology (a.k.a. General topology)


For multivariable calculus and linear algebra, most of the standard texts will do.  Note that multivariable calculus and linear algebra can be learned independently of one another (so it doesn't matter which one you learn first).
There are many good texts for learning real analysis, some of which are mentioned in the answers to this question.  It will be important that your real analysis education cover not only single-variable differentiation and integration, but also multivariable differentiation and integration as well.  In particular, it's important that you learn about the Inverse Function Theorem and Implicit Function Theorem.
Where point-set topology is concerned, some of it will hopefully be covered when you learn real analysis.  (Indeed, many analysis texts consider point-set topology to be a part of analysis.)  You will need to learn about metric spaces and topological spaces.  An excellent book for this is "Topology" (Munkres).
Finally, once you've gotten through all of this, I would say the text to use for manifold theory is "Introduction to Smooth Manifolds" (John Lee).  In fact, the book has an appendix at the end which gives a rapid treatment of each of these four subjects.  So, if you absolutely cannot contain your curiosity, this might be worth looking into.
Optional: One more helpful (but not necessary) pre-requisite is elementary differential geometry (or classical differential geometry), which is a beautiful topic to learn once you've finished with multivariable calculus.  My recommended textbook is "Elementary Differential Geometry" (Pressley).
