Open Problems and Questions about Geodesics — Question 25

Definitions and mathematical statement

Let $S$ be a closed surface of genus $h\ge 2$, as in the supplied source; closed means compact without boundary. The source does not explicitly specify orientability, so its genus convention requires confirmation.

For $r\in\{3,4,5,\ldots\}\cup\{\infty\}$, consider the following existence statement: there are a $C^r$ Riemannian metric $g$ on $S$ and a complete unit-speed geodesic $\gamma:{\mathbb{R}}\to S$ such that \[ \nabla^{g}_{\dot\gamma}\dot\gamma=0,\qquad g(\dot\gamma(t),\dot\gamma(t))=1,\qquad K_g(\gamma(t))=0\quad\text{for every }t\in{\mathbb{R}}, \] and \[ \nexists T>0\quad\forall t\in{\mathbb{R}}:\quad\gamma(t+T)=\gamma(t). \] Here $K_g:S\to{\mathbb{R}}$ is the Gaussian curvature and $\nabla^g$ is the Levi--Civita connection. The last condition expresses the usual interpretation of a non-closed geodesic as a nonperiodic geodesic; the source does not separately define that phrase.

Question 25 (attributed to Burns). Does such a metric and geodesic exist with $r=\infty$, or at least with $r=k$ for $k\ge 3$? The source does not specify whether its finite-regularity alternative asks for some such $k$, a prescribed $k$, or every such $k$. It also does not explicitly distinguish a question for an arbitrary prescribed surface from existence on some surface of genus at least two. These quantifier qualifications remain unresolved. No sign condition on Gaussian curvature away from the geodesic is supplied.

Short English statement

Burns asks whether a closed surface of genus at least two admits a smooth Riemannian metric with a nonperiodic geodesic along which the Gaussian curvature vanishes everywhere, allowing a finite-regularity alternative $C^k$, $k\ge 3$. The surface convention and some quantifiers require review. The registry labels the question open; that label is supplied metadata, not an independently verified assessment.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $800945$, source problem code catalogue entry, in the AMR Open Problem Lists collection. The record attributes the source list to Burns and Matveev and the question itself to Burns.
• Burns and Matveev, Open Problems and Questions about Geodesics (2019, as dated in the supplied record), Question 25. 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 reported source metadata, not a claim of independent access or verification.
• The supplied registry status is open. No independent literature assessment or source verification is asserted here.