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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 Length spectrum and Zoll surfaces of revolution and Length spectrum of spheres . 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 '12 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 '12 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 '12 at 22:12
If it is isospectral to the canonical metric then it is isometric. – Anton Petrunin Mar 9 '12 at 2:40
up vote 5 down vote accepted

The answer is positive; in fact any smooth manifold has two nonisometric metrics with conjugate geodesic flows. A construction is in C. Croke, B. Kleiner, Conjugacy and rigidity for manifolds with a parallel vector field. J. Differential Geom. 39(1994), 659–680.

The idea is quite simple: consider the followins two building blocks:

the round one: it is the round sphere without a small ball around the north pole.

the exotic one: take a Zoll metric of revolution such that it has the standard metric near the north pole, and cut out a small ball around the north pole (of the same radius as in the round building block).

Because the building blocks are isometric near boundary, if a manifold has a bump which is isometric to the round building block, then one can replace it by the exotic building block: the result is a smooth manifold and one can show that its geodesic flow is conjugate to the geodesic flow of the initial metric

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Hi Vladimir, I'll take a close look at this. Is the conjugacy symplectic (or more precisely, preserving the canonical $1$-form)? – alvarezpaiva Sep 3 '13 at 19:27
It has the same properties as the conjugation of the geodesic flows of the Zoll metrics. I believe it is symplectic and you probably should know better than me whether it preserves the canoncal 1-form. – Vladimir S Matveev Sep 4 '13 at 7:29
I was a bit confused about the construction because I didn't "see" the picture: a little wart on the manifold that passes from being perfectly round almost everywhere to being Zoll. I wonder if asking for a condition like "no waists" (no local minima of the energy functional on loops) is a reasonable condition to insure some sort of rigidity. – alvarezpaiva Sep 4 '13 at 8:19

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