Definition:Differentiable Structure

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

Let $C^k(\R^d)$ be the Space of k-differentiable functions $\R^d \to \R^d$.

Then a pre-differentiable structure of class $C^k$ $\mathscr F$ on $M$ is a collection of co-ordinate systems $\{(U_\alpha,\phi_\alpha) : \alpha \in A\}$ such that:


 * $(1): \quad \displaystyle \bigcup_{\alpha \in A} U_\alpha = M$


 * $(2): \quad \phi_\alpha \circ \phi_\beta^{-1}$ is $C^k(\R^d)$ for all $\alpha,\beta \in A$

If $\mathscr F$ is maximal with respect to condition $(2)$ in the sense that:


 * $(3): \quad$ If $(U,\phi)$ 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 $(U,\phi) \in \mathscr F$.

Then $\mathscr F$ is a differentiable structure of class $C^k$ on $M$.

If $\mathscr F$ satisfies $(1)$, $(3)$ and:


 * $(2a): \quad \phi_\alpha \circ \phi_\beta^{-1}$ is locally given by a convergent power series for all $\alpha,\beta \in A$

Then $\mathscr F$ is a differentiable structure of class $C^\infty$ on $M$.

If $M$ has dimension $2d$ for some integer $d$, then under the group isomorphism $\R^{2d} \simeq \C^d$ we can view a co-ordinate system $(U,\phi)$ as a map to complex $d$-space.

In this case, if $\mathscr F$ satisfies $(1)$, $(3)$ and:


 * $(2b): \quad \phi_\alpha \circ \phi_\beta^{-1}$ is holomorphic for all $\alpha,\beta \in A$

Then $\mathscr F$ is a complex analytic differentiable structure on $M$.