71 problems

A prime closed geodesic on a Finsler manifold is a closed geodesic that is not an iteration of another closed geodesic. For a Finsler metric on the sphere $S^n$, let $[x]$ denote t…

A Besse manifold is a Riemannian manifold all of whose geodesics are closed, and a prime geodesic is a closed geodesic that is not a multiple iterate of a shorter closed geodesic.…

A Finsler metric $F$ on the sphere $S^d$ is an irreversible Finsler metric defining the geodesic flow under consideration, and distinct closed geodesics are counted as geometricall…

Let $(M,g)$ be a compact Riemannian manifold without boundary, and call it a C-manifold if all of its geodesics are closed. Call $(M,g)$ Zoll if all of its closed geodesics have th…

Let $L$ be the length of the shortest closed geodesic on a Riemannian $2$-sphere, and let $area$ denote its area. The best known bound stated in the source is…

Let ${\mathcal T}^{\text{s}}_{1,1}$ denote the set of hyperbolic one-holed tori whose isometry group is isomorphic to $C_2\times S_3$. If $M\in {\mathcal T}^{\text{s}}_{1,1}$, and…

Let $\mathbf{M}$ be the modular torus, the hyperbolic once-punctured torus described in the paper. A pair of simple closed geodesics is a pair of simple geodesics on $\mathbf{M}$ w…

A once-punctured torus is a hyperbolic surface whose simple closed geodesics have a length spectrum, with multiplicity counting how many distinct simple closed geodesics have a giv…

Let $M$ be a complete, non-compact Riemannian manifold with locally convex ends. Chambers–Liokumovich–Nabutovsky–Rotman conjecture. Every such manifold admits a closed geodesic. Th…

Let $l\!k$ be a field and $X$ a simply connected finite CW complex. Write $LX=\operatorname{Map}(S^1,X)$ for the free loop space of $X$, and let $H^*(X;l\!k)$ denote its cohomology…

Erlandsson–Parlier conjecture. For every hyperbolic surface $X$,

Bounded-deviation strengthening. There exists $C>0$ such that if an orbit $\widetilde y,\dots,\widetilde f^{n_0}(\widetilde y)$ of $\widetilde f$ crosses $V_C(\widetilde\gamma)$ fo…

Let $F$ be a Finsler metric on $S^2$, and call a closed geodesic prime if it is not an iterate of a shorter closed geodesic. Two-or-infinity conjecture for Finsler metrics. Every F…

Let $n>2$, and let $(S^n,g)$ be an $n$-dimensional Riemannian sphere. A Besicovitch-type geodesic–hypersurface inequality asserts that there is a constant $C(n)>0$, depending only…

Let $n>2$, and let $(S^n,g)$ be an $n$-dimensional Riemannian sphere. A Besicovitch-type product inequality asserts that there is a constant $C(n)>0$, depending only on the dimensi…

Let $M$ be a closed Riemannian manifold of dimension at least two, and let a closed geodesic on $M$ mean a geodesic that is periodic under the geodesic flow. Conjecture on infinite…

Let $k$ be a positive integer, and let $S^{2k}$ or $S^{2k-1}$ be an even- or odd-dimensional sphere equipped with a Finsler metric. Katok–Anosov conjecture. Every Finsler metric on…

Let $s_0$ be the critical speed associated with a magnetic system $({\mathtt{g}},\sigma)$, and let $s$ satisfy $0<s<s_0$. A contractible closed $({\mathtt{g}},\sigma)$-geodesic is…

Let $M_k$ denote the maximal length of a closed geodesic with $k$ self-intersections on a hyperbolic surface, and let $\Gamma$ be a corkscrew geodesic on a thrice-punctured sphere.…

Let $(M,F)$ be a compact simply connected $n$-dimensional irreversible Finsler manifold. A prime closed geodesic is a closed geodesic that is not an iteration of another closed geo…

Let $S$ be a genus-zero surface and let the Turaev cobracket assign to each class of closed curve on $S$ its cobracket in the Goldman–Turaev Lie bialgebra. Chas's genus-zero cobrac…

Sparse equidistribution conjecture. The conclusions of the first theorem hold for every subgroup $H$ satisfying

Let $F$ be a Finsler metric on the two-sphere $S^2$, and let a closed geodesic be a periodic geodesic of $F$. Long's elliptic-geodesic conjecture. There exists at least one ellipti…

Let $F$ be a Finsler metric on the two-sphere $S^2$, and call a closed geodesic prime if it is not an iterate of a shorter closed geodesic. Long's hyperbolic-geodesic conjecture. I…

Let $F$ be an irreversible Finsler metric on the two-sphere $S^2$, and call a closed geodesic a periodic geodesic of $F$. Long's multiplicity conjecture. There must be either two o…