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].