Real/Complex Manifolds - Transition Maps I'm trying to understand how real/complex structure is imposed on a manifold, especially the likes of smooth manifolds. 
I can read the definitions and work with them, but I want to understand intuitively (i.e. geometrically) what is really going on. The lack in my understanding stems from grasping why imposing a condition on transition maps is "sufficient" to give the manifold the corresponding property. 
For example, a smooth manifold is a topological space with transition maps that are smooth. OK, fine. I imagine the sphere with a basic atlas. But what should I be visualizing in a more abstract setting when I think "the transition maps are smooth?" I've been trying to imagine a two-dimensional smooth manifold by folding a coordinate plane onto the surface of the manifold, and then sliding it over the surface ("changing coordinates") in a way that is "smooth." Is this a good way to imagine it, or does anyone have a good way to understand this concept? (Not only for smooth structure, but for any similar type of structure, though I assume the visualization would be the same.)
Thank you.
 A: You should maybe think of transition functions as "gluings". You're gluing smooth open sets together, so if you want the manifold you obtain to be smooth, the gluings should happen smoothly.
Here is the more useful way to see this when you actually work with an atlas. The argument is always the same: such property works when I look at it using some chart of the atlas, but what about if I use another chart? Then a transition function appears. Here is an example that I hope will make this clear:
So, let $M$ be a manifold with an atlas $(U_i, \varphi_i)$ and let $f : M \to \mathbb{R}$. Let $x$ be a point in $M$ and $(U_i, \varphi_i)$ a chart such that $x \in U_i$. We want to define smoothness of $f$ the point $x$ as follows:
Definition $f$ is said to be smooth at $x$ if $f \circ {\varphi_i}^{-1} : \varphi_i(U_i) \subset \mathbb{R}^n \to \mathbb{R}$ is smooth at $\varphi_i(x)$.
The question that comes to mind is: is this definition consistent? What happens if $x \in U_i$ and $x \in U_j$ for some $i \neq j$? I'll let you figure that out.
