  Let $(M,g)$ be a compact smooth $3$-dimensional Riemannian manifold without
boundary. It is proved that the energy-critical nonlinear Schr\"odinger
equation is globally well-posed for small initial data in $H^1(M)$, provided
that a certain tri-linear estimate for free solutions holds true. This estimate
is known to hold true on the sphere and tori in $3d$ and verified here in the
case $\mathbb{S}\times\mathbb{S}^2$. The necessity of a weak form of this
tri-linear estimate is also discussed.
