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The goal of this work is to prove global controllability and stabilizationproperties for the fractional Schr\"odinger equation on $d$-dimensional compactRiemannian manifolds without boundary $(M,g)$. To prove our main results we usetechniques of pseudo-differential calculus on manifolds. More precisely, byusing microlocal analysis, we are able to prove propagation of regularity whichtogether with the so-called Geometric Control Condition and Unique ContinuationProperty help us to prove global control results for the system underconsideration. As a main novelty this manuscript presents the relation betweenthe geometric control condition and the controllability for the fractionalSchr\"odinger equation.

The fractional Schr\"odinger equation on compact manifolds: Global controllability results