  We study quantitative stability results for different classes of Sobolev
inequalities on general compact Riemannian manifolds. We prove that, up to
constants depending on the manifold, a function that nearly saturates a
critical Sobolev inequality is quantitatively $W^{1,2}$-close to a non-empty
set of extremal functions, provided that the corresponding optimal Sobolev
constant satisfies a suitable strict bound. The case of sub-critical Sobolev
inequalities is also covered. Finally, we discuss degenerate phenomena in our
quantitative controls.
