Motivation for smooth structures

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

Charts

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

Recall that a chart is a homeomorphism of the form:

• [ilmath]\varphi:U\in\mathcal{J}\rightarrow U_\varphi\in\mathcal{R} [/ilmath]

(In this article [ilmath]A_\varphi[/ilmath] will denote the image of [ilmath]A[/ilmath] under the chart [ilmath]\varphi[/ilmath])

We can look at the map:

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

Let us suppose that:

• [ilmath]f\circ\varphi^{-1}:U_\varphi\rightarrow\mathbb{R} [/ilmath] as being differentiable.

Suppose also that we're given a second map:

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

We may now define:

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

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 [ilmath]f[/ilmath]

Smoothly compatible charts

If however we say that all charts in an atlas must be smoothly compatible then [ilmath]\varphi\circ\psi^{-1} [/ilmath] is differentiable (it is more than just differentiable, it is smooth!) and if we can define what it means to differentiate [ilmath]f[/ilmath] 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 [ilmath]f[/ilmath]"