## Franklin's Notes

### Ck manifold

A $C^k$ manifold modelled on an affine space $X$ (often $\mathbb R^n$) consists of a Hausdorff topological space $M$ together with a collection of open sets $U_a\subset M$ and corresponding maps $\phi_a:U_a\to X$ indexed by $a\in A$, with the following properties:

1. $\bigcup_{a\in A} U_a=M$, that is, the sets $U_a$ cover $M$.
2. Each $\phi_a$, when restricted to a function on its range $U_a\to \phi_a(U_a)$, defines a homeomorphism.
3. If $U_a\cap U_b\ne\varnothing$, then the composites $\phi_a\circ\phi_b^{-1}$ and $\phi_b\circ\phi_a^{-1}$ are $C^k$ maps $\to X$, where they are defined.

The pairs $(U_a,\phi_a)$ are called charts on $M$, and the collection of all of them (indexed by $a\in A$) is called an atlas. If a manifold is a $C^k$ manifold for all $k\in\mathbb N$, then it is called a $C^\infty$ manifold, or a smooth manifold.