Real-analyticity of 2-dimensional superintegrable metrics and solution of two Bolsinov-Kozlov-Fomenko conjectures
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2025
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| _version_ | 1866911565655048192 |
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| author | Matveev, Vladimir S. |
| author_facet | Matveev, Vladimir S. |
| contents | We study two-dimensional Riemannian metrics which are superintegrable in the class of polynomial in momenta integrals. The study is based on our main technical result, Theorem 3, which states that the Poisson bracket of two polynomial in momenta integrals is an algebraic function of
the integrals and of the Hamiltonian. We conjecture that two-dimensional superintegrable Riemannian metrics are necessary real-analytic in isothermal coordinate systems, and give arguments supporting this conjecture. Small modification of the arguments, discussed in the paper, provides a methods to construct new superintegrable systems. We prove a special case of the above conjecture which is sufficient to show that
the metrics constructed by K. Kiyohara in 2001, which admit irreducible polynomial in momenta integrals of arbitrary high degree $k$, are not superintegrable and in particular do not admit nontrivial polynomial in momenta integral of degree less than $k$. This result solves Conjectures (b) and (c) explicitly formulated in Bolsinov, KOzlov and Fomenko in 1995. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_18485 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Real-analyticity of 2-dimensional superintegrable metrics and solution of two Bolsinov-Kozlov-Fomenko conjectures Matveev, Vladimir S. Exactly Solvable and Integrable Systems Mathematical Physics Differential Geometry Dynamical Systems 37J35, 70H06 We study two-dimensional Riemannian metrics which are superintegrable in the class of polynomial in momenta integrals. The study is based on our main technical result, Theorem 3, which states that the Poisson bracket of two polynomial in momenta integrals is an algebraic function of the integrals and of the Hamiltonian. We conjecture that two-dimensional superintegrable Riemannian metrics are necessary real-analytic in isothermal coordinate systems, and give arguments supporting this conjecture. Small modification of the arguments, discussed in the paper, provides a methods to construct new superintegrable systems. We prove a special case of the above conjecture which is sufficient to show that the metrics constructed by K. Kiyohara in 2001, which admit irreducible polynomial in momenta integrals of arbitrary high degree $k$, are not superintegrable and in particular do not admit nontrivial polynomial in momenta integral of degree less than $k$. This result solves Conjectures (b) and (c) explicitly formulated in Bolsinov, KOzlov and Fomenko in 1995. |
| title | Real-analyticity of 2-dimensional superintegrable metrics and solution of two Bolsinov-Kozlov-Fomenko conjectures |
| topic | Exactly Solvable and Integrable Systems Mathematical Physics Differential Geometry Dynamical Systems 37J35, 70H06 |
| url | https://arxiv.org/abs/2501.18485 |