If $M$ and $N$ are manifolds with boundaries and $\{(U_a,f_a)\}$ and $\{(V_a,g_a)\}$ are their respectives $C^r$ atlas, why $\{(U_a \times V_b,f_a \times g_b)\}$ isn't an $C^r$ atlas for $M \times N$?
I tried to compose the maps and show that they are not $C ^ r$, but got no contradiction. It's probably simple, but I don't get it.
When $a\in \partial M$ and $b\in\partial N$, the map $f_a\times g_b$ sends a $C^r$-diffeomorphism of $U_a\times V_b$ onto a neighborhood of a corner in the product of two closed halfspaces. This is not what the charts on a manifold with boundary are supposed to do. They should map neighborhoods onto open subsets of $[0,\infty)\times \mathbb R^{d-1}$.