Smooth manifold

Remarks

The unit circle $S^1 \subset \R^2$ is a smooth manifold. Let $N = (0, 1)$ and $S = (0, -1)$, and let $\sigma_N \colon S^1 \setminus \set{N} \to \R$ and $\sigma_S \colon S^1 \setminus \set{S} \to \R$ be the stereographic projections, $\sigma_N(x, y) = x/(1 - y)$ and $\sigma_S(x, y) = x/(1 + y)$. On the overlap, $\sigma_N \sigma_S = x^2/(1 - y^2) = 1$, so the transition map is $t \mapsto 1/t$ on $\R \setminus \set{0}$, which is smooth with smooth inverse.

The charts $(\R, \id)$ and $(\R, t \mapsto t^3)$ on $\R$ are not smoothly compatible. The transition map $t \mapsto t^{1/3}$ is not differentiable at $0$. Each of them alone defines a smooth structure on $\R$, and the two structures are different, although the resulting manifolds are diffeomorphic.

Every smooth atlas is contained in a unique maximal one, so in practice a smooth manifold is given by any smooth atlas. Replacing $C^\infty$ by $C^k$ gives $C^k$ manifolds.