  We give an analytic description for the infinitesimal generator constructed
by Applebaum-Estrade for L\'evy flights on a broad class of closed Riemannian
manifolds including all negatively-curved manifolds, the flat torus and the
sphere. Various properties of the associated semigroup and the asymptotics of
the expected stopping time for L\'evy flight based random searches for small
targets, also known as the narrow capture problem, are then obtained using our
newfound understanding of the infinitesimal generator. Our study also relates
to the L\'evy flight foraging hypothesis in the field of biology as we compute
the expected time for finding a small target by using the L\'evy flight random
search. A similar calculation for Brownian motion on surfaces was done in
[arXiv:2209.12425].
