We prove that for any given integer $n\geq 2$ and $q\in [1,
n)$ there exists a constant $\epsilon= \epsilon(n,q)>0$ such
that any $n$-dimensional complete Riemannian manifold with
nonnegative Ricci curvature, in which the Sobolev inequality
¶
\[ \left(\int_M|f|^{\frac
{nq}{n-q}}\,dv\right)^{\frac{n-q}{nq}}\leq
(K(n,q)+\epsilon)\left(\int_M|\nabla f|^q
\,dv\right)^{\sfrac{1}{q}}, \,\,\forall f\in C_0^{\infty}(M)
\]
¶ holds with $K(n,q)$ the optimal constant of this inequality
in the $n$-dimensional Euclidean space $R^n$, is diffeomorphic
to~$R^n$.