  The goal of this work is to prove global controllability and stabilization
properties for the fractional Schr\"odinger equation on $d$-dimensional compact
Riemannian manifolds without boundary $(M,g)$. To prove our main results we use
techniques of pseudo-differential calculus on manifolds. More precisely, by
using microlocal analysis, we are able to prove propagation of regularity which
together with the so-called Geometric Control Condition and Unique Continuation
Property help us to prove global control results for the system under
consideration. As a main novelty this manuscript presents the relation between
the geometric control condition and the controllability for the fractional
Schr\"odinger equation.
