# Length spectrum and Zoll surfaces of revolution

The earlier MO question, "Length spectrum of spheres," asked if the length spectrum of closed geodesics determines the metric on $S^2$, and the answer was a clear No due to Zoll surfaces, all of whose geodesics are simple, closed, and of the same length. This made me wonder about the same question for metrics for surfaces of revolution, and I found a paper by Steve Zelditch, "Inverse Spectral Problem for Surfaces of Revolution" arXiv:math-ph/0002012v1, which seems to answer the question Yes:

Thus, if $(S^2,g)$ and $(S^2,h)$ are isospectral surfaces of revolution in $\cal{R}^*$, then $g$ is isometric to $h$.

The class $\cal{R}^*$ are rotationally invariant metrics of "simple type" and which "satisfy some generic non-degeneracy conditions." However, there are Zoll surfaces which are analytic surfaces of revolution, as explicitly stated, e.g., in Dan Jane's "The Ricci flow does not preserve the set of Zoll metrics" arXiv:0809.2722v1. So I've hit an apparent contradiction.

I don't fully understand Zelditch's conditions, and perhaps the contradiction disappears in that his conditions exclude Zoll surfaces. Or I may have some other elementary misunderstanding. If anyone can clarify the situation for me, I would be grateful. Thanks!

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The class $\mathcal{R}^*$ of surfaces considered by Zelditch excludes all Zoll surfaces of revolution, because of the "simple length spectrum" hypothesis (page 2) which is never satisfies for Zoll surfaces of revolution. For example it implies that the length $2L$ of "meridian" geodesics cannot be equal to the length of all other geodesics, while in the Zoll case all geodesics have the same length. – YangMills Mar 8 '12 at 4:40
Thank you, YangMills!! I have copied your comment into a CW answer for acceptance. – Joseph O'Rourke Mar 8 '12 at 11:17

The class $\cal{R}^*$ of surfaces considered by Zelditch excludes all Zoll surfaces of revolution, because of the "simple length spectrum" hypothesis (page 2) which is never satisfied for Zoll surfaces of revolution. For example it implies that the length $2L$ of "meridian" geodesics cannot be equal to the length of all other geodesics, while in the Zoll case all geodesics have the same length.