Distance in a Riemannian submanifold (compact)

Question

Let $M,S$ be compact connected Riemannian manifolds such that $S\subset M$ (injectively immersed). Denote $d_M$ and $d_S$ their respective Riemannian distances. Is $d_M$ restricted to $S\times S$ equivalent to $d_S$? I.e. is it true that $$A^{-1}d_M(x,y)\leqslant d_S(x,y)\leqslant Ad_M(x,y),\quad\forall x,y\in S,$$ for positive constant $A$?