= Orientability of factors of a product manifold
For nonempty <smooth manifolds> $M,N$, <orientability> of the <product manifold> $M\times N$ implies orientability of each factor. Fix $q\in N$ and an ordered <basis> of $T_qN$. Contract a positive top-degree <differential form> on the product with this fixed <basis> along $M\times\{q\}$. The result is a nowhere-zero top-degree form on $M$. Interchanging the factors proves the result for $N$.
Conversely, pulling back chosen orientation forms from the two factors and taking their <wedge product> gives an orientation of $M\times N$. The nonempty assumption is necessary if empty manifolds are allowed: a product with an empty factor is empty regardless of the other factor's <orientability>.
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