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 $A$?

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    $\begingroup$ You need to assume $S$ to be embedded in order to have the inequality on the right. $\endgroup$ – Onil90 May 9 at 20:32
  • $\begingroup$ If you only require your map to be immersed, there could be self intersection, i.e. there could be two different point $x, y \in S$ which are mapped to the same point in $M$. Therefore their distance in $S$ is positive, but obviously the distance of their image in $M$ is zero. $\endgroup$ – Onil90 May 10 at 17:20
  • $\begingroup$ I meant injectively immersed, the question is now edited. Also, I am assuming both $M$ and $S$ to be compact. $\endgroup$ – pipenauss May 10 at 17:49
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    $\begingroup$ Hint: Use the tubular neighborhood theorem and construct a $C^1$ (hence, Lipschitz) retraction from this neighborhood to $S$. $\endgroup$ – Moishe Kohan May 11 at 12:41

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