This is the question:
Let $M,N$ be Riemannian manifolds, such that the inclusion $i:M \to N$ is a isometric immersion. Give a example where the inequality $d_M > d_N$ may occur.
I thought about the immersion $i:\mathbb{S}^2 \to \mathbb{R}^3$, but I only have some geometrical insight, I don't know how to prove analytically. Please help me - thanks.
Let $M = S^1$ and $N = \mathbb{R}^2$. Immerse $M$ into $N$ in the "standard" way, so that $M$ is the unit circle in $N$. Give $S^1$ the pullback metric, so that this inclusion is an isometric immersion. (This is just the standard metric on the circle).
Then we have $d_{S^1}(x,y) > d_{\mathbb{R}^2}(x,y)$ for any $x,y\in S^1$ with $x\neq y$. This follows because a geodesic in the circle isn't straight, as seen by $\mathbb{R}^2$, hence not minimizing in $\mathbb{R}^2$.