The following content has been provided by the University of Erlangen-Nürnberg.

So good morning and welcome back. Today we'll turn to differentiable manifolds

and we pass from topological manifolds to differentiable manifolds

by actually removing charts from a topological atlas.

So that's the first step, 4.1, adding structure by refining the maximal topological atlas,

or the maximal C0 atlas.

So we saw before that if we have a topological manifold, so MO topological manifold,

that we can actually construct an atlas for this manifold,

and there is essentially only one maximal topological atlas,

because all the charts you can choose are automatically by virtue of the definition

of a topological manifold C0 compatible.

It was a fully redundant notion, an atlas was a fully redundant notion for a topological manifold.

But actually now we do something non-redundant, something non-trivial.

We consider a topological manifold MO and we call an atlas, or an atlas curly A,

is called a flower atlas if any two charts, ux and vy say,

that lie in the atlas are flower compatible.

So in other words, what we look at is that we have either the situation

that the two chart regions u and v, which are open subsets in the manifold,

do not intersect at all, then they are already flower compatible,

or if they have a non-zero intersection, then we require that if we map this intersection

employing the chart map x to x of u intersected v,

or alternatively employing the chart map y that is defined on v,

but then certainly also in the intersection y u v,

that if we consider this y after x inverse chart transition map,

and we recall this is the chart transition map,

then the observation already last time was that both x of u intersected v

and y of u intersected v by virtue of the definition of a topological manifold

and indeed charts in an atlas are subsets of Rd where Rd was the dimension of the manifold dim m we're looking at.

So the chart transition map is not between manifolds in general,

but between, it's a map from Rdm to Rdm,

and now the compact, we knew, we know from last time that in a topological manifold

such a chart transition map is always a homomorphism,

because x is a homomorphism, y is a homomorphism,

so the composition of y and the inverse of x is a homomorphism,

so they're always C0 compatible,

but now we don't want the charts transition maps to be pairwise C0 compatible,

but we want them to be flower compatible,

so the requirement is that the chart transition map must be flower.

Full stop.

Okay, so before you think I finally turned nuts, what is flower?

Can be.

So now there are various possibilities.

Well, the simplest possibility is flower can be C0.

So we had this before, we'd say the charts, the atlas is C0 compatible,

and the atlas is C0 as a map from Rd to Rd, we had this last time.

Another possibility is that you require a different structure.

The different structure would, another structure would be that it be Ck.

What does Ck mean?

Well, for manifolds we're about to define what Ck means,

but we push this down to a condition on the transition map,