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I'm studying a particular kind of curve evolution on Riemannian manifolds. It would help me to know the answer to the following kinda weird question:

Does there exist a closed Riemannian manifold $M$ and a pair of distinct closed geodesics $\gamma, \alpha$ in $M$ satisfying the follow properties?

(1) $\gamma$ and $\alpha$ are of the same length. (Call the length $l$.)

(2) $\gamma$ is isolated in the space of loops of length $l$.

(3) For any $\varepsilon>0$, there exists a path of loops, each of length between $l$ and $l+\varepsilon$, starting at $\gamma$ and ending at $\alpha$.

I suspect (and hope) that the answer is no.

Details: The loop space I'm working with is $C^0(S^1,M)$ with the compact-open topology. In (2), I mean that $\gamma$ has a neighborhood in the loop space in which the only loops of length $l$ are reparameterizations of $\gamma$. The path in (3) of course has to be a path of rectifiable loops.



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up vote 6 down vote accepted

Is it a homework problem?

Define the distance between curves as $$d(\gamma,\gamma')=\inf_h\sup_x|\gamma'(x)-\gamma\circ h(x)|,$$ where $h:\mathbb S^1\to\mathbb S^1$ is reparametrization. Fix small $\delta$ so that if $d(\gamma,\gamma')\le \delta$ and $\mathop{\rm length}\gamma=\mathop{\rm length}\gamma'$ then $\gamma'=\gamma\circ h$ for some $h$. Fix $\varepsilon>0$ a choose the path $\gamma_t$. Choose the smallest value $t$ such that $d(\gamma,\gamma_t)=\delta$ and set $\beta_\varepsilon=\gamma_t$. You may assume that $\beta_\varepsilon$ has unit-speed parametrization, in particular they all uniformly Lipschitz. Pass to partial limit as $\varepsilon\to 0$ and that is it.

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@Anton Petrunin: How exactly are you choosing the paths $\gamma_t$? Without specifying this, I don't understand how you can say "Choose the smallest value of $t$ such that $d(\gamma, \gamma_t)=\delta$." Also, it's been a while since I took a differential geometry course, but this questions seems much too hard for a standard HW exercise. – Aaron Trout Jul 26 '12 at 13:57
@Aaron, The path $\gamma_t$ is given in (3). – Anton Petrunin Jul 26 '12 at 14:25
@Anton Petrunin Thanks for your answer! It's not a homework problem, but you're right that I should have thought about the question a bit harder before asking on MO. I guess it's not as difficult as I expected. I don't know why your answer was downvoted. – Dmitri Gekhtman Jul 26 '12 at 15:02
Passing to the limit one gets a loop located at a distance $\delta$ from $\gamma$. By the semicontinuity it has a lenght less than or equal to $l$. But why is it exactly $l$? – Pietro Majer Jul 26 '12 at 20:54
@Pietro, you can always make it longer, say by composing with a zig-zag $\mathbb{S}^1\to\mathbb{S}^1$. – Anton Petrunin Jul 26 '12 at 23:13

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