Orientable products in $\mathcal {N}$

Abstract:

It is well known that the square $W \times W$ of any smooth closed manifold $W$ is cobordant to an orientable manifold. This note shows more specifically that a product $U \times V$ of smooth closed manifolds $U$ and $V$ is cobordant to an orientable manifold if and only if there is a smooth closed manifold $W$ such that $U$ and $V$ are both products of $W$ by orientable manifolds.