This paper deals with the theory of fractional Sobolev spaces on a compact Riemannian manifold (M, g). Our first main result shows that the fractional Sobolev spaces Ws,p(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W<<^>>{s,p}(M)$$\end{document} introduced by Guo et al. (Electron J Differ Equ 2018(156): 1-17, 2018) coincide with the classical Triebel-Lizorkin spaces (which in turn coincide with the Besov spaces). As an application, we study a non-local elliptic equation of the form LKu+h|u|p-2u=f|u|q-2u,\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\begin{aligned} {\mathcal {L}}_{{\mathcal {K}}}u + h|u |<<^>>{p-2}u = f|u |<<^>>{q-2}u, \end{aligned}$$\end{document}where the operator LKu\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathcal {L}}_{K} u$$\end{document} is an integro-differential operator a little more general than the fractional Laplacian, defined on Ws,p(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W<<^>>{s,p}(M)$$\end{document}. We use the Mountain Pass Theorem to show an existence result under a coercivity condition when we have a sub-critical non-linearity on the right-hand side of the Eq. (1). Our second main result is a Sobolev inequality in the critical range with an optimal constant for the fractional Sobolev spaces Ws,2(M)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$W<<^>>{s,2}(M)$$\end{document}. This inequality gives us a sufficient existence condition for (1) with p=2\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$p=2$$\end{document} and q=2*=2nn-2s\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$q=2<<^>>*=\frac{2n}{n-2s}$$\end{document} the fractional critical Sobolev exponent.