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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.

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.

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.

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.

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 metric 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 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.

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.