Open Problems and Questions about Geodesics — Question 43

Definitions and mathematical statement

Let $M$ be a smooth compact manifold of dimension $3$ without boundary, equipped with a reversible Finsler metric $F$. Thus $F$ is smooth away from the zero section, positively homogeneous of degree one in each tangent space, and strongly convex, and reversibility means \[ F(x,v)=F(x,-v)\qquad(x\in M,\ v\in T_xM). \] Assume that $F$ has positive flag curvature: $K_F(x,y,P)>0$ for every $x\in M$, every nonzero $y\in T_xM$, and every two-dimensional plane $P\subset T_xM$ containing $y$.

Let $N\subset M$ be a closed two-dimensional totally geodesic submanifold. Here closed means compact without boundary, and totally geodesic means that geodesics of the induced Finsler metric on $N$ are also geodesics of $(M,F)$. The draft uses the usual embedded-submanifold convention; the supplied source does not explicitly specify whether immersed submanifolds are also intended.

A closed geodesic is a nonconstant periodic $F$-geodesic $\gamma:{\mathbb{R}}/T{\mathbb{Z}}\to M$, with $T>0$. Is it true that, for every $(M,F)$ and $N$ satisfying these hypotheses and every such $\gamma$, \[ \gamma({\mathbb{R}}/T{\mathbb{Z}})\cap N\ne\varnothing \quad\text{or}\quad \gamma({\mathbb{R}}/T{\mathbb{Z}})\subseteq N\,? \] The second alternative already implies the first under the ordinary set-theoretic meaning of intersection. Both alternatives are retained from the source; no transversality requirement is asserted.

Short English statement

Must every nonconstant closed geodesic on a closed reversible Finsler three-manifold of positive flag curvature meet a given closed totally geodesic surface, including the possibility that it lies entirely in that surface?

Sources

• Local UnsolvedMath registry, permanent ID $800962$, source problem code catalogue entry, title: Open Problems and Questions about Geodesics --- Question 43. The supplied record attributes the question to Burns and Matveev, with proposed year 2019.
• Burns, Matveev, Open Problems and Questions about Geodesics (2019), Question 43. 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 independently verified access or source verification.
• The registry labels the problem open. This is the supplied status label, not an independently verified assessment of the literature.