Open Problems and Questions about Geodesics — Question 7

Definitions and mathematical statement

The supplied question concerns Kropina metrics but does not specify their defining conventions. Under the usual smooth, globally defined convention, let $M$ be a closed smooth manifold (compact and without boundary), let $g$ be a smooth Riemannian metric, and let $\beta$ be a smooth nowhere-vanishing one-form. Set \[ \mathcal A=\{(x,v)\in TM:\beta_x(v)>0\},\qquad F(x,v)=\frac{g_x(v,v)}{\beta_x(v)}\quad ((x,v)\in\mathcal A). \] Thus $F=\alpha^2/\beta$, where $\alpha(x,v)=\sqrt{g_x(v,v)}$; this is the Kropina metric on its admissible cone $\mathcal A$.

An admissible closed geodesic means a nonconstant smooth map $\gamma:{\mathbb{R}}/T{\mathbb{Z}}\to M$, for some $T>0$, satisfying $\beta_{\gamma(t)}(\dot\gamma(t))>0$ for every $t$, and stationary for the energy \[ E_F(\gamma)=\frac12\int_0^T F(\gamma(t),\dot\gamma(t))^2\,{\,\mathrm{d}} t \] under every smooth periodic variation through admissible curves, with $T$ fixed.

Question (Burns--Matveev, Question 7, supplied attribution 2019). Does every Kropina metric on a closed manifold admit an admissible closed geodesic? Under the convention above, this asks whether, for every such $(M,g,\beta)$, there exist $T>0$ and $\gamma$ with the stated properties.

Unresolved qualification. The supplied excerpt does not define Kropina metrics or state their regularity, dimension, or global domain conventions. The displayed formulation uses the smooth convention with nowhere-vanishing $\beta$ and domain $\beta>0$; its agreement with the intended source convention requires review. In particular, the excerpt does not settle whether data with zeros of $\beta$, and hence empty admissible fibers there, are included.

Short English statement

Burns and Matveev ask whether every Kropina metric on a compact manifold without boundary has a nonconstant periodic geodesic whose velocity always lies in the metric's admissible cone. The supplied record labels this question open; that label is not an independently verified status assessment. The precise convention for Kropina metrics requires source review.

Sources

• Local UnsolvedMath registry, permanent ID $800927$, source code catalogue entry, title Open Problems and Questions about Geodesics --- Question 7. The supplied record attributes the question to Burns and Matveev, with proposed year 2019, and labels it open.
• Burns and Matveev, Open Problems and Questions about Geodesics (2019, as identified in the supplied source list), Question 7. Supplied source URL: https://arxiv.org/abs/1308.5417. The provenance records access on 2026-10-16 and extraction by arxiv_latex; these are supplied provenance claims, not independent verification performed for this draft.
• The supplied statement cites [CMMM2019] for relations between Kropina metrics and the so-called chains in CR geometry, offered as motivation for recent interest. No bibliographic entry or URL for this reference is supplied.