  The objective of this paper is twofold. First, we conduct a careful study of
various functional inequalities involving the fractional Laplacian operators,
including nonlocal Sobolev-Poincar\'e, Nash, Super Poincar\'e and logarithmic
Sobolev type inequalities, on complete Riemannian manifolds satisfying some
mild geometric assumptions. Second, based on the derived nonlocal functional
inequalities, we analyze the asymptotic behavior of the solution to the
fractional porous medium equation, $\partial_t u +(-\Delta)^\sigma (|u|^{m-1}u
)=0 $ with $m>0$ and $\sigma\in (0,1)$. In addition, we establish the global
well-posedness of the equation on an arbitrary complete Riemannian manifold.
