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. 


Thanks,


Dmitri 



A strange question about closed geodesics on a closed manifold
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.


