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 $N$.