Ck compatibility for atlases and differentiable structures

Question

Recently i started watching prof. Frederic Schuller's lectures on the geometrical anatomy of theoretical physics. I am currently learning about manifolds and atlases and charts and while i understand that for two charts $(U, \varphi)$ and $(V, \psi)$ to be $C^k$ compatible the transition map $\varphi(\psi)$ and its inverse have to be $C^k$ i don't understand how this notion is lifted to atlases.

In my notes i have that

a $C^k$ atlas that is $C^0$ compatible with a $C^0$ atlas that defines a topological manifold is said to determine a differentiable structure on that manifold

I don't really understand what does it mean for two atlases to be $C^0$ compatible, do all the charts in an atlas $\mathfrak{A}$ have to be $C^0$ compatible with the charts in $\mathfrak{B}$ ? I don't get it. Also how do these two atlases determine a differentiable structure

Also i have that

The $C^k$ equivalence classes of such atlases are the $C^k$ differentiable structures, each one is determined by a unique maximal atlas which is the union of all atlases in the equivalence class

I don't fully understand this, can someone explain this to me?

Answer 1

I do not know how Schuller defines a topological manifold, but the usual way is to say that a topological manifold of dimension $n$ is a topological space $M$ such that

1. $M$ is locally Euclidean which means that each $p \in M$ has an open neighborhood which is homeomorphic to an open subset of $\mathbb R^n$.

2. $M$ is Hausdorff.

3. $M$ has a countable basis.

The essential property is 1. The technical properties 2. and 3. exclude certain "pathologies".

A chart on $M$ is any homeomorphism $\phi : U \to U'$ from an open $U \subset M$ to an open $U' \subset \mathbb R^n$. An atlas $\mathfrak A$ on $M$ is any collection of charts whose domains cover $M$. The transition functions between two charts in such an atlas are in general only continuous maps (i.e. of class $C^0$). Since the inverse of a transition function is also a transition function, we see that each transition function is a homeomorphism between open subsets of $\mathbb R^n$.

Thus we can define a $C^0$-atlas on $M$ to be any atlas on $M$ because its transition functions are of class $C^0$. Similarly a $C^k$-atlas is one with transition functions are of class $C^k$. Clearly each $C^k$-atlas is also a $C^l$-atlas for $l = 0,\ldots, k-1$.

Two $C^k$-atlases $\mathfrak A, \mathfrak B$ are called $C^k$-compatible if their union $\mathfrak A \cup \mathfrak B$ is again a $C^k$-atlas. This specializes to $k = 0$, but it does not mean much in this case: The union of any two ($C^0$-)atlases is always a ($C^0$-)atlas.

Writing

a $C^k$ atlas that is $C^0$-compatible with a $C^0$-atlas that defines a topological manifold is said to determine a differentiable structure on that manifold

does therefore not mean very much: The union of a $C^k$-atlases and a $C^0$-atlas is always a $C^0$-atlas.

Finally, call two $C^k$-atlases $\mathfrak A, \mathfrak B$ equivalent if they are $C^k$-compatible. For each equivalence class of $C^k$-atlases the union of all its members is a $C^k$-atlas. In other words, each equivalence class contains a unique maximal $C^k$-atlas. Such a maximal $C^k$-atlas is usually called a $C^k$ differentiable structure on $M$.