Abstract: Let $\mathcal{L}$ be the Laplace-Beltrami operator. On an $n$-dimensional $ (n\geq 2)$, complete, noncompact Riemannian manifold $\mathbb{M}$, we prove that if $0<\alpha<1, s>\alpha/2$ and $f \in H^s(\mathbb{M})$, then the fractional Schrödinger propagator ${\rm e}^{{\rm i}t|\mathcal{L}|^{\alpha/2}}(f)(x)\rightarrow f(x)$ a.e. as $t\rightarrow0$. In addition, for when $\mathbb{M}$ is a Lie group, the rate of the convergence is also studied. These results are a non-trivial extension of results on Euclidean spaces and compact manifolds.

Key words: Schrödinger propagator, noncompact manifolds, spectra