What is meant by "rigidity of a geometric structure"? I often heard description like "complex structures are much more rigid than smooth structures", but I have never managed to understand this notion of rigidity.
When I asked what "rigidity" exactly means, I often got explanations like "the space of smooth functions is much larger than the space of holomorphic functions." I know that for compact complex manifolds, the only holomorphic functions are constants, but why this gives us a sense of rigidity?
Another heuristic explanation goes like this: "there is not much direction allowed to deform a complex structure, so that it remains a complex structure after deformed." This is even more confusing. What is meant by "not much direction to deform"?
After all, how can we measure the "rigidity" of a certain geometric structure in general? Could someone explain this notion of rigidity in more detail?
 A: In general, I think "more rigid" just means "subject to more constraints." That's why you're getting explanations about classes of functions containing each other, and examples of special subcategories of larger categories. It also ties in with user72694's illustration in the comments (which is probably the most on-target considering your description) that because the more constrained requirements of analytic functions force them to behave more specially than smooth functions.
This can be illustrated a bit in transformational geometry, where you view the space as a set acted on by transformations.
So for example, triangles in affine geometry ($GL(\Bbb R^n)$ operating on $\Bbb R^n$) are not very rigid: you can transform any triangle into another with an affine transformation.
But Euclidean geometry has less transformations, and therefore less flexibility and more rigidity. That's why there are many very different classes of triangles in Euclidean geometry.
Going to the extreme, you could let your transformation group be maximally rigid and make it just the identity transformation. Within this geometry, a triangle is not even the same as a translated version of itself. Any nontrivial movement at all of a triangle would change it into a completely different triangle: that's a pretty rigid triangle!

Here's another example I think is in the same vein. Let's consider "rigidity" of functions from $\Bbb Z\to \Bbb Q$. 
If it's just a function, you've got a lot of choice of how to make a map. Any random function at all works.
You can make the function more rigid by requiring it to preserve order, then you have much less choice.
Or for another thing, you could require that the function be additive. If it were additive and you had to send $1$ to $1$, then you'd have absolutely no choice at all in your map: you'd have to use the identity injection.
More constraints on the function are leading to less choice (more rigidity) about the nature of the function.
