Entanglement and products

We consider a sequence of quantum states, ${\rho }_N$, associated with a submanifold $\mathrm{\Lambda }$ of product of two integral compact K"ahler manifolds. We show that, when $\mathrm{\Lambda }$ is a product submanifold, then, in the semiclassical limit, these states are not entangled. We discuss whether geometry of $\mathrm{\Lambda }$ (specifically $\mathrm{\Lambda }$ being a product submanifold) has any relationship with entanglement properties of ${\rho }_N$.