Open Problems and Questions about Geodesics — Conjecture 60 ([Bo-Ko-Fo1995])

Definitions and mathematical statement

Let $T^2={\mathbb{R}}^2/{\mathbb{Z}}^2$. Under the usual smooth, globally defined interpretation, let $g$ be a smooth Riemannian metric on $T^2$. A polynomial first integral of degree exactly three in the velocities is a smooth function $F:TT^2\to{\mathbb{R}}$ of the form \[ F(x,v)=A_0(x)+A_1(x)(v)+A_2(x)(v,v)+A_3(x)(v,v,v), \] where $A_k$ is a smooth symmetric covariant $k$-tensor field and $A_3\not\equiv0$, such that, for every affinely parametrized $g$-geodesic $\gamma:I\to T^2$, \[ \frac{{\,\mathrm{d}}}{{\,\mathrm{d}} t}F\left(\gamma(t),\dot\gamma(t)\right)=0 \qquad(t\in I). \] A nonzero Killing vector field is a smooth vector field $X\not\equiv0$ satisfying $\mathcal L_Xg=0$; equivalently, with $\nabla$ the Levi--Civita connection, \[ g(\nabla_YX,Z)+g(Y,\nabla_ZX)=0 \] for all smooth vector fields $Y,Z$.

Conjecture 60, under these conventions. For every such metric $g$, if its geodesic flow admits a polynomial first integral of degree exactly three in the velocities, then $g$ admits a nonzero Killing vector field.

Unresolved qualifications. The supplied wording does not specify smoothness, global versus local integrals, or whether the cubic integral must be homogeneous. The formulation above explicitly uses smooth global data and allows lower-degree terms. For the homogeneous reading, require $A_0=A_1=A_2=0$. These conventions require source review. The word "nonzero" is also implicit rather than explicit in the supplied conclusion; without it, the zero vector field would satisfy the conclusion for every metric.

Short English statement

The supplied conjecture says that a Riemannian metric on the two-dimensional torus whose geodesic flow has a cubic polynomial conserved quantity must have a nonzero infinitesimal isometry. The precise regularity, globality, and homogeneity conventions require review.

Sources

• Local UnsolvedMath registry, supplied complete source record: permanent ID $800979$, exact source code catalogue entry; upstream revision d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Burns and Matveev, Open Problems and Questions about Geodesics (2019, as identified by the supplied record), Conjecture 60. Supplied source URL: https://arxiv.org/abs/1308.5417. The supplied provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not independently verified access.
• The source title and background associate the conjecture with the reference label [Bo-Ko-Fo1995]. No corresponding bibliographic entry is supplied, so its authorship and publication details remain unresolved. The registry lists Burns and Matveev as proposers and 2019 as the proposed year; this does not resolve the attribution represented by that reference label.
• The registry labels the problem "open". This is a supplied status label, not an independently verified assessment of the literature.