# Distance in a Riemannian submanifold (compact)

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 $$A$$?

• You need to assume $S$ to be embedded in order to have the inequality on the right. – Onil90 May 9 at 20:32
• 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. – Onil90 May 10 at 17:20
• I meant injectively immersed, the question is now edited. Also, I am assuming both $M$ and $S$ to be compact. – pipenauss May 10 at 17:49
• Hint: Use the tubular neighborhood theorem and construct a $C^1$ (hence, Lipschitz) retraction from this neighborhood to $S$. – Moishe Kohan May 11 at 12:41