Smooth reversible counterexamples and an exact symmetrization diagnostic

Definitions and mathematical statement

Let $M$ be a smooth closed orientable surface of genus at least two. A smooth strongly convex Finsler metric $F$ is positive and positively homogeneous of degree one on $TM\setminus0$, with positive-definite fiber Hessian of $F^2/2$. The source asks: if every unparameterized $F_1$-geodesic is an $F_2$-geodesic, must there exist $c>0$ and a smooth closed real 1-form $\beta$ on $M$ such that \[ F_1(x,v)=cF_2(x,v)+\beta_x(v)\qquad((x,v)\in TM)? \] The original does not specify whether the unparameterized curves retain orientation. The proposed higher-dimensional extension concerns closed manifolds admitting a hyperbolic metric, not necessarily metrics $F_i$ of negative curvature themselves.

The supplied research aid reports a negative answer in the smooth category: on every such orientable surface there are smooth strongly convex reversible Finsler metrics with the same oriented unparameterized geodesics, yet with no representation $F_1=cF_2+\beta$ for any $c>0$ or any 1-form $\beta$, even without requiring closedness. Reversibility means $F_i(x,-v)=F_i(x,v)$, so these examples also address the unoriented convention. The reported construction uses Lang's smooth Crofton construction and handle attachments in an open region where the metrics agree. The record provides neither a detailed construction nor a full reference. It also attributes rigidity in the real-analytic category to Lang, without full theorem hypotheses. These are supplied solved-in-literature reports, not independently verified results.

For clarity, the aid gives an exact algebraic diagnostic. Define \[ F_i^+(x,v)=\tfrac12(F_i(x,v)+F_i(x,-v)),\qquad F_i^-(x,v)=\tfrac12(F_i(x,v)-F_i(x,-v)). \] For a fixed $c>0$, an identity $F_1=cF_2+\beta$ with a smooth 1-form holds exactly when $F_1^+=cF_2^+$ and $F_1^--cF_2^-$ is fiberwise linear. That odd difference is then the unique $\beta$; the proposed AIM normal form additionally requires ${\,\mathrm{d}}\beta=0$. For reversible pairs it forces $\beta=0$. For Randers metrics $F_i=\alpha_i+\theta_i$, where $\alpha_i$ are Riemannian norms and $|\theta_i|_{\alpha_i}<1$, the normal form is equivalent to the existence of one constant $c>0$ with \[ \alpha_1=c\alpha_2,\qquad {\,\mathrm{d}}\theta_1=c\,{\,\mathrm{d}}\theta_2, \] in which case $\beta=\theta_1-c\theta_2$.

The original passage reports an affirmative Riemannian higher-dimensional statement for manifolds admitting hyperbolic metrics, citing Matveev, and describes an integral of the first geodesic flow constructed from the second metric. It notes that in the Riemannian surface case the integral is quadratic and yields rigidity through Kolokoltsov. These special results do not establish the unrestricted smooth Finsler assertion.

Short English statement

Must projectively equivalent Finsler metrics on a higher-genus surface differ only by a constant scale and a closed 1-form? The supplied aid reports smooth reversible counterexamples and an analytic rigidity contrast. Its even/odd decomposition gives a precise test for the proposed normal form.

Sources

• Question 10.1.3, AIM geodesics problem list: https://aimath.org/pastworkshops/geodesicsproblems.pdf. Supplied references are [Matveev2003], [Mat-Top1998], [Kolokoltsov1983] and [Mat-Top2000]; full entries are absent.
• The research aid names Lang's smooth Crofton construction and real-analytic theorem without title, year, identifier or complete hypotheses.
• Local registry data/unsolvedmath/problems.json, permanent ID 20001930, code catalogue entry; canonical aim-geometry-notes.json, notes[267], source item 10.1; updated 14 August 2026. Clean statement null; statement status reconstructed unverified.
• Attempt 1: counterexample, problem status at run refuted, rigor label proved, novelty confidence low; current solved and solved-in-literature labels. The aid is machine generated, not peer reviewed, and requires independent verification before citation.