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.