Smooth manifold

1. $\mathbb{R}^n$ with the smooth structure generated by 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.

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.

Smooth manifolds are the objects of the category $\mathbf{Diff}$, whose morphisms are smooth maps. Its categorical isomorphisms are exactly the diffeomorphisms. Thus a coordinate chart does not make the whole manifold isomorphic to Euclidean space; rather, its coordinate map is an isomorphism between an open submanifold and an open submanifold of $\mathbb R^n$. See coordinate charts as isomorphisms in $\mathbf{Diff}$.