Open Problems and Questions about Geodesics — Question 46 (Babenko)

Definitions and mathematical statement

Let $S^2$ be the two-dimensional sphere, and let $g_{\mathrm{rd}}$ denote its round metric of constant Gaussian curvature $1$, so that $\operatorname{Area}_{g_{\mathrm{rd}}}(S^2)=4\pi$. Write $\mathcal M$ for the space of smooth Riemannian metrics on $S^2$; smoothness is the conventional interpretation used in this draft, since the supplied statement gives no explicit regularity class. For $g\in\mathcal M$, a nonconstant closed geodesic is a nonconstant periodic map $\gamma:{\mathbb{R}}/{\mathbb{Z}}\to S^2$ satisfying $\nabla^g_{\dot\gamma}\dot\gamma=0$. Its length is \[ L_g(\gamma)=\int_0^1\sqrt{g_{\gamma(t)}(\dot\gamma(t),\dot\gamma(t))}\,{\,\mathrm{d}} t. \] The source asks whether every area-$4\pi$ metric sufficiently close to $g_{\mathrm{rd}}$ admits such a geodesic of length at most $2\pi$, with equality allowed. More explicitly, once a topology $\mathcal T$ on $\mathcal M$ is specified, the proposed assertion is \[ \exists\,U\in\mathcal T\quad g_{\mathrm{rd}}\in U\quad\text{and}\quad \forall g\in U,\qquad \operatorname{Area}_g(S^2)=4\pi \ \Longrightarrow\ \exists\,\gamma\text{ a nonconstant closed }g\text{-geodesic with }L_g(\gamma)\leq2\pi. \] The supplied record does not specify $\mathcal T$, so the meaning of sufficiently close remains unresolved; this draft does not select a topology.

The source also describes the question as asking whether $\sigma\geq1/\pi$ near the round metric, but does not define $\sigma$. If $\ell(g)$ denotes the length of a shortest nonconstant closed $g$-geodesic and the intended convention is \[ \sigma(g)=\frac{\operatorname{Area}_g(S^2)}{\ell(g)^2}, \] then, under the area normalization, the stated bound becomes \[ \ell(g)\leq2\pi\quad\Longleftrightarrow\quad\sigma(g)\geq\frac1\pi. \] This interpretation of $\sigma$ is conditional and requires confirmation from the source context.

Short English statement

The question attributed in the supplied record to Babenko asks whether every Riemannian metric on $S^2$ of area $4\pi$, sufficiently close to the unit round metric, has a nonconstant closed geodesic of length at most $2\pi$. The topology of closeness and the definition of the source's quantity $\sigma$ are not supplied.

Sources

• Local UnsolvedMath registry, permanent ID $800965$, exact source problem code catalogue entry; supplied title: Open Problems and Questions about Geodesics --- Question 46 (Babenko). The registry lists Burns and Matveev as proposers and gives the year 2019; the background records attribution/context to Babenko.
• Burns and Matveev, Open Problems and Questions about Geodesics (2019, as identified by the supplied record), Question 46. Supplied source URL: https://arxiv.org/abs/1308.5417. The record reports access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance details, not a claim of independent source access or verification.
• The supplied registry labels the problem open. This is a supplied classification, not an independently verified assessment of the literature or present mathematical status.