# Manifold Explained

Is there a good explanation of a manifold on the web somewhere? The wikipedia article isn't really working for me. I was actually hoping for a whiteboard lecture on youtube, but can't find one.

My math experience is calculus through differential equations, twenty five years ago. I also have some computer science related math; discrete structures and numerical methods.

My problem with the Wikipedia article is that I just can't visualize what they're saying.

Any sources (not just web based) are appreciated.

Edit 2

I stumbled upon the Wikipedia article while trying to brush up on my math. I had been talking to my daughter about primes, we wandered there from her 4th grade homework. We ventured as far as Mersene primes and perfect numbers. It was at this point that I knew I had to brush up.

Somehow I chased a link to Riemann manifolds and this was completely new to me.

Having looked at some of the suggested material from the answers, the Wikipedia article makes a lot more sense. In fact I'm trying to figure out why I found it unclear, maybe it can be improved.

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You need to give some details of what you want to know, and/or of your background. As thinks stand, pointing you to Warner's book on manifolds might seem like a sensible answer... – Mariano Suárez-Alvarez Dec 8 '10 at 3:05
Can you be more specific about what's not working for you about the Wikipedia article? – Qiaochu Yuan Dec 8 '10 at 3:10
Web only? , we can recommend some interesting books – Bryan Yocks Dec 8 '10 at 3:16
Could you give us some details on what exactly you're looking for? I'm guessing you aren't interested in a formal definition. But the Wikipedia page is already fairly light. Do you want more of a cartoon explanation? FYI, here is a related thread: math.stackexchange.com/questions/8808/… – Ryan Budney Dec 8 '10 at 5:56
One reason why one might be interested in manifolds is that generic level-sets of smooth functions are manifolds. So if you know some quantity is conserved for solutions to an ODE, you know that generically the dynamics is happening on a manifold. So you could use properties of those manifolds. Is that the kind of thing you're interested in? The theorem I'm quoting is called "Sard's theorem". – Ryan Budney Dec 8 '10 at 5:58

This isn't in any way a thorough or rigorous introduction to manifolds, but as far as visualization it's quite nice: Weeks' The Shape of Space.

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I liked Lee's book on Manifolds. You can find the first chapter here on his website linked below.

http://www.math.washington.edu/~lee/Books/Manifolds/c1.pdf

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thanks, that's a good start. – uncle brad Dec 8 '10 at 15:01
The link above is an old one that doesn't work any more. Try this: math.washington.edu/~lee/Books/ITM/c01.pdf – Jack Lee Jul 4 '14 at 17:03