  If one thinks of a Riemannian metric, $g_1$, analogously as the gradient of
the corresponding distance function, $d_1$, with respect to a background
Riemannian metric, $g_0$, then a natural question arises as to whether a
corresponding theory of Sobolev inequalities exists between the Riemannian
metric and its distance function. In this paper we study the sub-critical case
$p < \frac{m}{2}$ and show a Sobolev inequality exists where an
$L^{\frac{p}{2}}$ bound on a Riemannian metric implies an $L^q$ bound on its
corresponding distance function. We then use this result to state a convergence
theorem and show how this theorem can be useful to prove geometric stability
results by proving a version of Gromov's conjecture for tori with almost
non-negative scalar curvature in the conformal case. Examples are given to show
that the hypotheses of the main theorems are necessary.
