  This paper deals with fractional Sobolev spaces on a compact Riemannian
manifold. We prove a Sobolev inequality in the critical range with an optimal
constant for these fractional Sobolev spaces. We use this result to study the
existence of a non-trivial solution for equations driven by a non-local
integro-differential operator $\mathcal{L}_{\mathcal{K}}$ with critical
non-linearity.
