arXiv research

The paper studies the shortest closed multi-geodesics on hyperbolic surfaces as their genus grows.

The paper finds shortest geodesic bounds on orbifolds with diameter limits.

Shortest non-simple closed geodesics on hyperbolic surfaces found.

Shortest geodesic on curved spheres is no longer than 3 times the diameter.

Study shortest non-simple geodesics on 2-orbifolds, finding unique shortest curve.

The paper finds bounds on shortest dense curves on surfaces.

Improved bounds on shortest geodesics with self-intersections on hyperbolic surfaces.

The paper improves bounds on the shortest closed geodesic length on surfaces.

Study shows shortest geodesic length on certain manifolds is limited by volume, diameter, and cover elements.

We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.

We give a metric characterization of the Euclidean sphere in terms of the lower bound of the sectional curvature and the length of the shortest closed geodesics.

Study bounds the length of shortest periodic geodesics on certain curved spaces.

We compute the number of systoles, the shortest simple closed geodesics and 2-systoles, the second shortest simple closed geodesics on hyperbolic surfaces homeomorphic to once-punctured torus and four-punctured sphere.

Sharp bounds found on shortest geodesic on punctured spheres.

Study on shortest geodesics crossing multiple times on hyperbolic surfaces with cusps.

In every conformal class of Finsler (or Riemannian) metrics on a closed manifold there exists a residual subset of Finsler metrics, such that, with respect to the residual Finsler metrics, in any non-trivial homotopy class of free loops there is precisely one shortest geodesic loop.

The study finds a bound on the shortest geodesics in hyperbolic 3-manifolds.

Study intersection numbers, lengths, and shortest geodesics on hyperbolic surfaces.

Upper bound established for the length of shortest closed geodesics in hyperbolic link complements.

Study shortest geodesics on flat cone spheres with conical singularities.

This note is about a type of quantitative density of closed geodesics on closed hyperbolic surfaces. The main results are upper bounds on the length of the shortest closed geodesic that $\varepsilon$-fills the surface.

In this paper, we show that for any closed 4-dimensional simply-connected Riemannian manifold $M$ with Ricci curvature $|Ric|\leq 3$, volume $vol(M)>v>0$, and diameter $diam(M)<D$, the length of a shortest closed geodesic is bounded by a function $F(v,D)$ which only depends on $v$ and $D$. The proofs of our result are …

Recently, F. Balacheff proved that the Calabi-Croke sphere made of two flat 1-unit-side equilateral triangles glued along their boundaries is a local extremum for the length of the shortest closed geodesic among the Riemannian spheres with conical singularities of fixed area. We give an alternative proof of this theore…

We show that the volume of any Riemannian metric on a three sphere is bounded below by the length of the shortest closed curve that links its antipodal image. In particular, the volume is bounded below by the minimum of the length of the shortest closed geodesic and the minimal distance between antipodal points.

In this paper we study 1/k geodesics, those closed geodesics that minimize on all subintervals of length $L/k$, where $L$ is the length of the geodesic. We develop new techniques to study the minimizing properties of these curves on doubled polygons, and demonstrate a sequence of doubled polygons whose closed geodesics…

The study of random surfaces reveals asymptotic lengths of separating geodesics.

The length of shortest non-simple geodesics grows logarithmically with surface genus.

We prove an upper bound for the number of shortest closed geodesics in a closed hyperbolic manifold of any dimension in terms of its volume and systole, generalizing a theorem of Parlier for surfaces. We also obtain bounds on the number of primitive closed geodesics with length in a given interval that are uniform for …

Improved bounds on geodesic lengths in Riemannian surfaces.

Study geodesics and shortest arcs on Lie groups with specific metrics.

Study geodesics and shortest arcs on Lie groups with specific metrics.

In this paper, we prove that on every Finsler $n$-sphere $(S^n, F)$ with reversibility $λ$ satisfying $F^2<(\frac{λ+1}λ)^2g_0$ and $l(S^n, F)\ge π(1+\frac{1}λ)$, there always exist at least $n$ prime closed geodesics without self-intersections, where $g_0$ is the standard Riemannian metric on $S^n$ with constant curvat…

We prove that if the unit codisc bundle of a closed Riemannian manifold embeds symplectically into a symplectic cylinder of radius one then the length of the shortest nontrivial closed geodesic is at most half the area of the unit disc.

There are many equivalent definitions of Riemannian geodesics. They are naturally generalised to sub-Riemannian manifold, but become non-equivalent. We give a review of different definitions of geodesics of a sub-Riemannian manifold and interrelation between them. We recall three variational definitions of geodesics as…

The study shows how many diameter directions in Besse manifolds relate to Blaschke manifolds.

Let x and y be two (not necessarily distinct) points on a closed Riemannian manifold M of dimension n. According to a celebrated theorem by J.P. Serre there exist infinitely many geodesics between x and y. The length of the shortest of these geodesics is obviously less than the diameter of M. But what can be said about…

New bounds on shortest geodesic loops on a sphere.

Any finite configuration of curves with minimal intersections on a surface is a configuration of shortest geodesics for some Riemannian metric on the surface. The metric can be chosen to make the lengths of these geodesics equal to the number of intersections along them.

Study shows shortest periodic geodesic on hyperbolic orbisphere complements figure-eight knot.

Paper finds shortest geodesic paths on hyperbolic surfaces.

Paper shows mapping class group-equivariant Teichmüller space deformation to Thurston spine.

We bound the dimension of the fiber of a Riemannian submersion from a positively curved manifold in terms of the dimension of the base of the submersion and either its conjugate radius or the length of its shortest closed geodesic.

Study finds shortest geodesic loops on Stiefel manifold and calculates its injectivity radius.

We introduce a pair of isospectral but non-isometric compact flat 3-manifolds called Tetra (a tetracosm) and Didi (a didicosm). The closed geodesics of Tetra and Didi are very different. Where Tetra has two quarter-twisting geodesics of the shortest length, Didi has four half-twisting geodesics. Nevertheless, these spa…

Explicit bounds found for shortest orthogeodesics and volumes of hyperbolic manifolds.

Study on shortest arcs on hyperbolic surfaces with boundary.

We study non-reversible Finsler metrics with constant flag curvature 1 on S^2 and show that the geodesic flow of every such metric is conjugate to that of one of Katok's examples, which form a 1-parameter family. In particular, the length of the shortest closed geodesic is a complete invariant of the geodesic flow. We …

The action of the mapping class group of the thrice-punctured projective plane on its $\mathrm{GL}(2,\mathbb{C})$ character variety produces an algorithm for generating the simple length spectra of quasi-Fuchsian thrice-punctured projective planes. We apply this algorithm to quasi-Fuchsian representations of the corres…