Euclidean Product of Differentiable Manifolds

Theorem

Theorem. Let $M$ and $N$ be smooth manifolds with dimensions $m$ and $n$. THen $M\times N$ is a smooth manifold with a ${\mathcal{C}}^{\infty }$ structure determined by coordinate neighborhood of the form $\{U\times V,\phi \times \psi \}$, where $(U,\phi )$ and $(V,\psi )$ are coordinate neighborhoods on $M$ and $N$, respectively, and $\phi \times \psi :{\mathbb{R}}^m\times {\mathbb{R}}^n\to {\mathbb{R}}^{m+n}$.

References

@Boo75 Chapter 3 Theorem 1.7