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. The first deals with compact manifolds with or without boundary, while the second approach treats the case of riemannian manifolds without boundary whose Ricci curvature is uniformly bounded below.