Pointwise convergence of solution to Schrodinger equation on manifolds

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 Schrödinger equation converges pointwisely to its initial data. Assume the initial data is in $H^α(M)$. For Hyperbolic Space, standard Sphere and the 2 dimensional Torus, we prove that $α>\frac{1}{2}$ is enough. For general compact manifolds, due to lacking of local smoothing effect, it is hard to beat the bound $α>1$ from interpolation. We managed to go below 1 for dimension $\leq 3$. The more interesting thing is that, for 1 dimensional compact manifold, $α>\frac{1}{3}$ is sufficient.

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