  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.
