Entanglement and products

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