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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.

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1 Answer 1

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 $\mathbb{R}^2$.

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This is more easier then my example, thanks. One could imagine if $i:\mathbb{S}^n \to \mathbb{R}^{n+1}$ is indeed a example too. –  Jr. Jul 7 '12 at 21:07
    
@Jr. It is an example for all $n$. One could show this using the $n=1$ case because $d_{S^1} = d_{S^n}$ for any great circle subset of $S^n$ with its usual metric. –  Jason DeVito Jul 7 '12 at 21:11
    
I can "see" your statement geometrically, but I don't understand it when $n \geq 3$ as I can't "see" $\mathbb{R}^4$ –  Jr. Jul 7 '12 at 21:16

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