I have been asked to start a thread on the basics of differential geometry/topology i.e. What the hell is a manifold?

I will limit my talks to surfaces (i.e. 2 dimensional manifolds) as most of them are easy to handle, the theory is very rich and most "mathematical gems" people recognise come from this group - like the mobius band or klein bottle.

Also, i can skip a lot of the machinery at the beginning and introduce a surface as a level set and not as an abstract maximal collection of atlases on some set

Well first we need a definition -

Alevel setof a smooth function f:U -> R where U is some open subset of the R<sup>3</sup> is the set f<sup>-1</sup>(c) = {(x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>) in R<sup>3</sup> : f(x<sub>1</sub>, x<sub>2</sub>, x<sub>3</sub>) = c}

An example is worth a thousand words (or one picture apparently) so consider the function f(x, y, z) = x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> which is "nice" everywhere except at the origin which happens to be an open subset of R<sup>3</sup>. Now what is the level surface f<sup>-1</sup>(4)?

Well it is the set of all points (x, y, z) such that x<sup>2</sup> + y<sup>2</sup> + z<sup>2</sup> = 4 i.e. the sphere of radius 2 centred at the origin.

Any questions about level sets?