Open Problems and Questions about Geodesics — Problem 41

Definitions and mathematical statement

Let $S^2$ be the two-dimensional sphere. Under the standard smooth, strongly convex convention, a Finsler metric is a function $F:TS^2\to[0,\infty)$, positive on nonzero tangent vectors, smooth away from the zero section, positively homogeneous of degree one in each tangent space, and such that the fibrewise Hessian of $\tfrac12F^2$ is positive definite away from the zero section. The supplied record does not specify regularity conventions or require reversibility, namely $F(x,v)=F(x,-v)$.

Write $K_F(x,v)$ for the flag curvature at a nonzero tangent vector $v\in T_xS^2$. A metric has constant flag curvature $K$ if $K_F(x,v)=K$ for every such $(x,v)$. Use the standard convention that $F$ is Zoll if every unit-speed geodesic is periodic and all have the same least positive period. A Killing vector field is a smooth vector field $X$ whose local flow $\phi_t$ preserves $F$: \[ F\left(\phi_t(x),d\phi_t|_x(v)\right)=F(x,v) \] whenever the flow is defined. The phrase "no Killing vector field" is understood to mean no nonzero Killing vector field.

Construction problem, literal final-sentence formulation. Construct a Finsler metric $F$ on $S^2$ and a constant $K\in{\mathbb{R}}$ such that \[ K_F(x,v)=K\quad\text{for every }x\in S^2,\ v\in T_xS^2\setminus\{0\}, \qquad F\text{ is not Zoll}, \] and every Killing vector field of $F$ is identically zero.

Unresolved scope. The preceding source paragraph explicitly discusses flag curvature $1$, whereas the final construction request says only "constant curvature." Thus the intended requirement may be the more specific condition $K=1$; the supplied excerpt does not resolve this distinction. It also refers to conditions $(i)$ and $(ii)$ without providing them, so any intended additional requirements from those conditions cannot be stated precisely.

The supplied background asserts that metrics satisfying $(i)$ necessarily admit a Killing vector field and are well understood; that applying the Zermelo transformation to them produces non-Zoll metrics satisfying $(ii)$; and that every metric of flag curvature $1$ on $S^2$ satisfying $(ii)$ and admitting a Killing vector field arises by that procedure. It further reports that no other non-Zoll examples satisfying $(ii)$ were known. These are supplied source qualifications, not independently verified assertions or a current literature assessment.

Short English statement

Construct a constant flag curvature Finsler metric on $S^2$ that is not Zoll and has no nonzero Killing vector field. The surrounding text suggests curvature $1$, but the final request does not specify its value. Referenced conditions $(i)$ and $(ii)$ are missing. This is a statement-only draft requiring human mathematical review.

Sources

• Local UnsolvedMath registry, permanent ID $800960$, exact source problem code catalogue entry, title "Open Problems and Questions about Geodesics --- Problem 41." The supplied record attributes the problem to Burns and Matveev, with proposed year 2019, and labels it open; that label is supplied metadata, not an independently verified status assessment. Upstream revision: d371ee917ac9647fbb06cd6fc529deec6f3dda82.
• Burns and Matveev, Open Problems and Questions about Geodesics (2019, as identified in the supplied record), Problem 41: https://arxiv.org/abs/1308.5417. The background records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance statements, not a claim of independent source access or verification.
• The supplied background cites [Fou-Mat2018] for the Zermelo transformation. No full bibliographic entry or link for this reference is supplied, so its identity and precise supporting result remain unresolved.