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Given a Riemannian manifold $(M,g)$, and an embedded sub-manifold $N\rightarrow M$ on which we equip the induced metric. For simplicity, let's suppose that $M$ and $N$ are both complete Riemannian manifolds, and Riemannian distances are defined. We denote them by $d_M$ and $d_N$ respectively. The question is, under what conditions do we have the following relation: $$ d_N(x_1, x_2) \leq k d_M(x_1, x_2), \, \forall x_1, x_2 \in N $$ for some positive constant $k$.

When $N$ is the compact, the question is easy. So we just need to consider unbounded $N$.

dj wu
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  • Can you give an example when $d_N(x, y) >d_M(x,y)$? – Berci Feb 11 '20 at 21:56
  • Taking $N$ as the circle on the plane. – dj wu Feb 11 '20 at 22:52
  • In the present form, this question is unanswerable: Besides compactness, or $f$ which is a diffeomorphism whose inverse has a uniform bound on $||Df^{-1}(x)||$, there are no general conditions which will guarantee such a Lipschitz estimate. – Moishe Kohan Feb 13 '20 at 19:05
  • Thanks Kohan. Yes, I realized that this is not a good question. However, there is a very special case, namely, when $N$ is totally geodesic, the condition is then verified. – dj wu Feb 15 '20 at 08:36

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