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: In (2) and (3), I didn't specify which loops I'm considering or how I'm topologizing the loop space. I guess for now I'll say that my loop space is $C^0(S^1,M)$ with the compact-open topology. In (3) I'm talking about a path of rectifiable loops. Actually, it suffices for my purposes to work with $H^1(S^1,M)$ with its natural Hilbert manifold topology (see e.g. [1]), but I think the answer will be the same.

Thanks,

Dmitri

[1] W. Klingenberg, Lectures on Closed Geodesics, Springer, 1978.