  We prove a bilinear $L^2(\R^d) \times L^2(\R^d) \to L^2(\R^{d+1})$ estimate
for a pair of oscillatory integral operators with different asymptotic
parameters and phase functions satisfying a transversality condition. This is
then used to prove a bilinear refinement to Strichartz estimates on closed
manifolds, similar to that on $\R^d$, but at a relevant semi-classical scale.
These estimates will be employed elsewhere to prove global well-posedness below
$H^1$ for the cubic nonlinear Schr\"odinger equation on closed surfaces.
