Integral representation for fractional Laplace-Beltrami operators

In this paper we provide an integral representation of the fractional Laplace-Beltrami operator for general riemannian manifolds which has several interesting applications. We give two different proofs, in two different scenarios, of essentially the same result. One of them deals with compact manifolds with or without boundary, while the other approach treats the case of riemannian manifolds without boundary whose Ricci curvature is uniformly bounded below.