Open Problems and Questions about Geodesics — Question 38

Definitions and mathematical statement

For the standard Riemannian interpretation, let $(M,g)$ be a manifold with a smooth Riemannian metric. A closed geodesic means a nonconstant smooth periodic geodesic: a map $\gamma:{\mathbb{R}}/L{\mathbb{Z}}\to M$, for some $L>0$, satisfying $\nabla^g_{\dot\gamma}\dot\gamma=0$. Define the union of their images by \[ \mathcal C(M,g)=\bigcup_{\substack{L>0,\ \gamma:{\mathbb{R}}/L{\mathbb{Z}}\to M\\ \gamma\text{ a nonconstant }g\text{-geodesic}}}\gamma({\mathbb{R}}/L{\mathbb{Z}}). \] Let $\mathcal K$ denote the class of manifolds and metrics intended by the source. The question is whether \[ \forall (M,g)\in\mathcal K,\qquad \overline{\mathcal C(M,g)}^{\,M}=M. \] Equivalently, must every nonempty open subset $U\subseteq M$ meet the image of a nonconstant closed $g$-geodesic? Density is asked in $M$ itself, not in its unit tangent bundle $SM$, and no genericity assumption on the metric is imposed.

The supplied excerpt does not identify $\mathcal K$. In particular, it does not specify compactness, completeness, boundary assumptions or boundary conditions, dimension, or connectedness. The Riemannian and smooth conventions above make the terminology explicit but require confirmation against the surrounding source. Consequently, this is a conditional formulation with an unresolved ambient class, not a claim that the question ranges over all smooth Riemannian manifolds.

Short English statement

Must the images of nonconstant closed geodesics be dense in the manifold itself for every metric allowed by the source setting, without assuming the metric is generic? The supplied excerpt omits the ambient hypotheses needed to specify that setting.

Sources

• Local UnsolvedMath registry, permanent ID $800958$, exact source problem code catalogue entry, title Open Problems and Questions about Geodesics --- Question 38. The record attributes the question to Burns and Matveev, gives proposed year 2019, and labels its status as open; this is a supplied registry label, not an independently verified mathematical assessment.
• Burns and Matveev, Open Problems and Questions about Geodesics (2019, as identified in the supplied record), Question 38, https://arxiv.org/abs/1308.5417. The supplied provenance reports access on 2026-10-16 and extraction by arxiv_latex. These are supplied provenance details; no independent source visit or verification is claimed.