Open Problems in Integrable Systems — Superintegrable systems

Definitions and mathematical statement

Shared notation.

$\mathbb N=\{1,2,\ldots\}$, and $\mathbb Z,\mathbb Q,\mathbb R,\mathbb C$ denote the integers, rationals, reals and complex numbers. Any other conventions are specified locally.

For a smooth Riemannian metric $g$ on $M$, a natural Hamiltonian on the canonical symplectic cotangent bundle $T^*M$ is \[H(x,p)=\tfrac12\sum_{i,j}g^{ij}(x)p_ip_j+U(x),\] with smooth potential $U$ and inverse metric coefficients $g^{ij}$. A polynomial integral of degree $d$ is a smooth function $F(x,p)$ polynomial of degree $d$ in the cotangent variables $p$, satisfying $\{H,F\}=0$. Following the source, integrals which are functions of $H$ alone are trivial. A geodesic flow is the case $U=0$. A metric is locally conformally flat if near each point there are coordinates and a positive smooth function $a$ with $g=a\sum_i(dx_i)^2$. The geodesic flow on an $n$-manifold is maximally polynomially superintegrable if it has $2n-1$ global first integrals, including $H$, polynomial in momenta, with linearly independent differentials almost everywhere on $T^*M$; these extra integrals are not required to mutually Poisson commute.

Statement recorded in the supplied source.

For a sphere $S^n$ with $n>2$, does there exist a smooth non-conformally-flat Riemannian metric whose geodesic flow is maximally polynomially superintegrable? [S2]

Short English statement

Can a non-conformally-flat sphere metric have the maximum possible number of independent polynomial geodesic integrals?

Sources

• S1 ULAM.AI and the UnsolvedMath Contributors, UnsolvedMath: A Curated Collection of Open Mathematics Problems, record 2100308 (AMR-020-0308). CC BY 4.0. https://huggingface.co/datasets/ulamai/UnsolvedMath.
• S2 A. Bolsinov, V. S. Matveev, E. Miranda and S. Tabachnikov, Open Problems, Questions, and Challenges in Finite-Dimensional Integrable Systems (2018), arXiv:1804.03737, Question 3.11. https://arxiv.org/abs/1804.03737.