Motivation for smooth structures

Suppose we have a topological manifold $M$ and a function $f:M\rightarrow\mathbb{R}$ which is continuous. Notice that the notion of continuity on such a map is easy! However suppose we want to differentiate $f$, what does this mean?

Charts

Here, $(M,\mathcal{J})$ is a topological $n$-manifold and $\mathcal{R}$ denotes the topology on $\mathbb{R}^n$, thus we can say $A\in\mathcal{R}$ as a short hand for "$A$ is open in $\mathbb{R}^n$", $\psi$ and $\varphi$ are charts from some atlas.

Recall that a chart is a homeomorphism of the form:

• $\varphi:U\in\mathcal{J}\rightarrow U_\varphi\in\mathcal{R}$

(In this article $A_\varphi$ will denote the image of $A$ under the chart $\varphi$)

We can look at the map:

• $f\circ\varphi^{-1}:U_\varphi\mathop{\subseteq}_\text{open}\mathbb{R}^n\rightarrow\mathbb{R}$ as being differentiable (as now we can look at directional derivatives very easily)

Let us suppose that:

• $f\circ\varphi^{-1}:U_\varphi\rightarrow\mathbb{R}$ as being differentiable.

Suppose also that we're given a second map:

• $\psi:V\in\mathcal{J}\rightarrow V_\psi\in\mathcal{R}$ where $V\cap U\ne\emptyset$ (that is to say the charts overlap, and thus a transition map exists)

We may now define:

• $f\circ\varphi^{-1}\circ(\varphi\circ\psi^{-1}):\psi(U\cap V)\rightarrow\mathbb{R}$, notice that $\underbrace{f\circ\varphi^{-1}}_\text{differentiable}\circ\underbrace{\varphi\circ\psi^{-1} }_\text{need not be}=f\circ\psi^{-1}$

So even on a patch of the manifold where both charts are defined there is no reason to assume that they both allow us (some notion of differentiability we can use) to differentiate $f$

Smoothly compatible charts

If however we say that all charts in an atlas must be smoothly compatible then $\varphi\circ\psi^{-1}$ is differentiable (it is more than just differentiable, it is smooth!) and if we can define what it means to differentiate $f$ we can do so for any charts we like.

This motivates the idea for a smooth atlas, the 'differentiable structure' talked about is exactly this structure on the manifold so that we can talk of "differentiating $f$"