Open Problems and Questions about Geodesics — Problem 44 (Benjamin Schmidt)

Definitions and mathematical statement

Let $(M,g)$ be an oriented closed Riemannian surface of genus at least $1$, and let $d_g$ be its Riemannian distance. A nontrivial free homotopy class $\Gamma$ is a free homotopy class of loops in $M$ that does not contain a constant loop; no primitivity assumption is stated. A closed geodesic written as $\gamma:{\mathbb{R}}\to M$ is understood through its periodic extension: for some $T>0$, $\gamma(t+T)=\gamma(t)$, and the loop $s\mapsto\gamma(Ts)$, with $s\in{\mathbb{R}}/{\mathbb{Z}}$, represents $\Gamma$.

The supplied source attributes the following surface assertion to Gaidukov: for every such $(M,g)$, every nontrivial free homotopy class $\Gamma$, and every $p\in M$, there exist a closed geodesic $\gamma:{\mathbb{R}}\to M$ representing $\Gamma$ and a ray $\beta:[0,\infty)\to M$ such that \[ \beta(0)=p, \qquad \lim_{t\to\infty}d_g\left(\beta(t),\gamma(t)\right)=0. \] This is convergence at matching parameter values, rather than merely convergence of the distance to the image of $\gamma$.

Requested generalization. Generalize this surface assertion to higher dimensions.

Unresolved formulation qualifications. The supplied record does not specify the higher-dimensional manifold class, additional geometric or topological hypotheses, or whether the conclusion should remain exactly the displayed convergence assertion. Thus it does not determine a single precise higher-dimensional conjecture. It also does not explicitly define its convention for a ray or specify the speed normalization and parametrization of the two curves. If a geodesic ray is intended, global distance minimization in $M$ cannot be required for an infinite unit-speed ray on a closed manifold, since its diameter is finite. The intended convention must be checked in the cited source before a fully specified generalization is formulated.

Short English statement

The supplied surface theorem says that, from every point of an oriented closed Riemannian surface of positive genus, a ray approaches a periodically parametrized closed geodesic in any prescribed nontrivial free homotopy class at matching times. The problem asks for a higher-dimensional generalization, but leaves its hypotheses and the precise ray convention unspecified.

Sources

• Supplied local UnsolvedMath registry record, permanent ID $800963$, exact source code catalogue entry, in the AMR Open Problem Lists collection. The registry labels the problem open; that label is supplied metadata, not an independently verified mathematical or literature assessment.
• Burns and Matveev, Open Problems and Questions about Geodesics (2019), Problem 44, as identified by the supplied record: https://arxiv.org/abs/1308.5417. The record reports access on 2026-10-16 and extraction from arXiv LaTeX; these provenance details have not been independently verified here. The registry lists Burns and Matveev as proposers and gives Benjamin Schmidt as attribution/context.
• Gaidukov, source reference label [Gaidukov1966], cited in the supplied statement for the surface theorem. The same statement also spells the name Gajdukov. No full bibliographic entry is supplied.
• Source reference label [Bia-Pol1986]. The supplied statement says this reference explains how Gaidukov's results follow easily from the classical results cited as [Morse1924] and [Hedlund1932]. Full bibliographic entries for these three references are not supplied.