## Length spectrum for Riemannian metrics in the projective plane

Are there (known) examples of non-isometric Riemannian metrics on the projective plane that have the same length spectrum ?

This question is related to MO questions http://mathoverflow.net/questions/90530 and http://mathoverflow.net/questions/89882/ . There Zoll surfaces appear as counterexamples of the analogous question on $S^2$ (and spoil all the fun), so maybe one should concentrate on the projective plane where the only Zoll Riemannian metric is the the canonical metric.

Remark. Notice that there are tons of very nice reversible Finsler Zoll metrics on the projective plane. Indeed, here is the Busemann recipe to cook up smooth reversible Finsler metrics on $RP^2$ such that all geodesics are projective lines :

1. Take a smooth strictly positive measure on the unit sphere in $\mathbb{R}^3$ that is invariant under the antipodal map.
2. If $x$ and $y$ are distinct, non-antipodal points, let $X$ and $Y$ denote the great circles obtained by intersecting the sphere with the subspaces orthogonal to $x$ and $y$.
3. The union of $X$ and $Y$ cuts the sphere into four connected components.
4. Define the distance between $x$ and $y$ as the measure of the smallest of these components.
5. Voilà, you have a metric on the sphere that being invariant under the antipodal map projects down to a metric on the projective plane. It is easy to see that projective lines are geodesics and not too hard to see that it is Finsler.
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 Would you please remind what is length spectrum? – Dmitri Mar 8 2012 at 21:30 @Dmitri: My understanding is that it is the multiset of the lengths of the closed geodesics: list all the lengths of those geodesics, sort them, and record how many times a length is repeated. A variation is the simple length spectrum, the multiset of the lengths of the simple closed geodesics. – Joseph O'Rourke Mar 8 2012 at 21:56 There seem to be various notions that go under the name "length spectrum". The simplest, that I've also seen called the length set is simply the set of lengths of periodic geodesics. One may want to record multiplicities and then you get the multiset that Joseph mentioned in his remark. For my question, I'd be quite happy to know if there are examples of non-isometric Riemannian metrics on the projective plane with the same length set. – alvarezpaiva Mar 8 2012 at 22:12 If it is isospectral to the canonical metric then it is isometric. – Anton Petrunin Mar 9 2012 at 2:40