  Let $(M^n,g)$ be a Riemannian manifold without boundary. We study the amount
of initial regularity is required so that the solution to free Schr\"{o}dinger
equation converges pointwisely to its initial data. Assume the initial data is
in $H^\alpha(M)$. For Hyperbolic Space, standard Sphere and the 2 dimensional
Torus, we prove that $\alpha>\frac{1}{2}$ is enough. For general compact
manifolds, due to lacking of local smoothing effect, it is hard to beat the
bound $\alpha>1$ from interpolation. We managed to go below 1 for dimension
$\leq 3$. The more interesting thing is that, for 1 dimensional compact
manifold, $\alpha>\frac{1}{3}$ is sufficient.
