Smooth manifold

Definition

Let $n\in\mathbb{N}$. An $n$-dimensional smooth manifold is a pair $(M,\mathcal A)$ such that

• $M$ is an $n$-dimensional topological manifold (Hausdorff, second countable, and every $p\in M$ has a neighborhood homeomorphic to an open subset of $\mathbb{R}^n$), and
• $\mathcal A$ is a maximal smooth atlas of coordinate charts on $M$.

Maximal means: if $(U,\varphi)$ is a chart on $M$ whose transition maps with every chart in $\mathcal A$ are smooth, then $(U,\varphi)\in\mathcal A$. Any (not-necessarily-maximal) smooth atlas determines a unique maximal one by adjoining all charts smoothly compatible with it.

Once a smooth structure is fixed, one can define intrinsic objects such as the tangent space at a point , the tangent bundle , and differential forms with the exterior derivative . Maps between smooth manifolds are compared using charts; see smooth maps .

Examples

1. $\mathbb{R}^n$ with the single global chart $(\mathbb{R}^n,\mathrm{id})$ is a smooth manifold; its maximal atlas consists of all charts whose coordinate changes are smooth.
2. The sphere $S^n\subset\mathbb{R}^{n+1}$ becomes a smooth manifold using the two stereographic projection charts from the north and south poles; their overlap transition map is smooth, so they generate a maximal smooth atlas.
3. Any Lie group is, by definition, a smooth manifold for which multiplication and inversion are smooth maps .