Intuition for the role of diffeomorphisms

I understand the general role of isomorphisms in mathematics. If two groups are isomorphic, they are indistinguishable by group-theoretic means. If two topological spaces are homeomorphic, they are indistinguishable by topological means. And so on.

However, I'm not quite sure if I understand the role of diffeomorphisms fully. In Differential Geometry of Curves & Surfaces, do Carmo writes that "from the point of view of differentiability, two diffeomorphic surfaces are indistinguishable." However, it is possible that two surfaces - say, the unit sphere and a sphere of radius 2 - are diffeomorphic although there are important differences: Because the diffeomorphism between them is a not an isometry, their inner geometry is different. If a curve is moved by the diffeomorphism from one of the spheres to the other, it changes its length.

So, the inner geometry is not necessarily preserved. But what is preserved? What does a diffeomorphism do that a homeomorphism doesn't do? (FWIW, I'm more interested in the intuition than in a technical description.)

Diffeomorphisms are the isomorphisms in the category of smooth manifolds, while isometries are the isomorphisms in the category of Riemannian manifolds.

That is to say, diffeomorphisms are under no obligation to preserve the extra structure (metric, and all the geometry that comes with it) that the manifolds might have. They do preserve the differentiable structures, in the sense that if $\varphi:M\to N$ is a diffeomorphism, then $g:N\to \Bbb R$ is smooth if and only if $g\circ \varphi :M\to \Bbb R$ is smooth.

You're getting at the fundamental difference between differential geometry and differential topology. When we are considering manifolds (surfaces are a $2$d case) we don't only care about the set, we care about all the structure. So, what structure are we preserving?

• A homeomorphism is an isomorphism of spaces equipped with topologies
• A diffeomorphism is an isomorphism of spaces equipped with smooth structures
• An isometry is an isomorphism of spaces equipped with metric (or Riemannian) structures

When we have a "new" category, like when we pass from (Abelian) groups to rings, we require that the maps between the objects have properties that preserve the structure. For instance, a(n) (Abelian) group homomorphism preserves the binary operation: $\phi:A\to B$ has $\phi(x+y)=\phi(x)+\phi(y)$. A ring homomorphism has the additional requirement that $\phi(xy)=\phi(x)\phi(y)$ for any pair $x,y\in A$.

If we have a pair of topological spaces $(X,\mathcal{T}_X)$ and $(Y,\mathcal{T}_Y)$, they have the data of a set of points and a topology on them (the $\mathcal{T}$ guys above). Then, a continuous map is a map which preserves the topological structure in a suitable sense. So, what exactly is a smooth manifold? A smooth manifold is a triple $(M,\mathcal{T}_M, \mathcal{A}_M$), where $\mathcal{A}_M$ is the datum of a smooth atlas. So, given a pair of smooth manifolds $M$ and $N$ (suppressing the triple notation) we require our "smooth" map $f:M\to N$ to preserve the topological structure and the smooth structure. Thus $f$ should be continuous with some additional properties.

Now, a smooth structure amounts to saying what functions are called smooth on the space. And a smooth map $f:M\to N$ is defined in such a way that if $g\in \mathscr{C}^\infty(N)$, then $f^*(g)=g\circ f\in \mathscr{C}^\infty(M)$ is also smooth. In this sense, a smooth function defines a map $f^*: \mathscr{C}^\infty(N)\to \mathscr{C}^\infty(M)$. More particularly, for any open subset $U\subseteq N$, by continuity $f^{-1}(U)\subseteq M$ is open and we get a map $f^*: \mathscr{C}^\infty(U)\to \mathscr{C}^\infty(f^{-1}(U))$ which relates smooth functions on one open set to smooth functions on the other.

Then, if we agree that this is the correct notion of a smooth map, the notion of diffeomorphism follows naturally.