Definition:Differentiable Structure

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

Let $\mathscr F$ be a pre-differentiable structure of class $\mathcal C^k$ on $M$.

Then $\mathscr F$ is a differentiable structure of class $\mathcal C^k$ iff:


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