Definition:Differentiable Structure

Definition
Let $M$ be a locally Euclidean space of dimension $d$.

An differentiable structure of class $\mathcal C^k$ on $M$ is an equivalence class of the set of atlases of class $\mathcal C^k$ on $M$ under the equivalence relation of compatibility.

Equivalent Definitions
We may define a differentiable structure of class $\mathcal C^k$ on $M$ to be an atlas $\mathscr F$ of class $\mathcal C^k$ on $M$ that is maximal in the following sense:


 * Whenever $\left({U, \phi}\right)$ is a chart of $M$ such that $\phi \circ \phi_\alpha^{-1}$ and $\phi_\alpha \circ \phi^{-1}$ are $C^k$ for all $\alpha \in A$, then $\left({U, \phi}\right) \in \mathscr F$.

Another equivalence definition is as follows: let $\mathcal G$ be an equivalence class of the set of atlases of class $\mathcal C^k$ on $M$ under the equivalence relation of compatibility.

Then a differentiable structure of class $\mathcal C^k$ on $M$ is a maximal element of $\mathcal G$, partially ordered by inclusion.

Also see

 * Compatibility of Atlases is Equivalence Relation
 * Inclusion is Partial Order on Sets
 * Equivalence of Definitions of Differentiable Structure