arXiv research

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

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

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 shortest geodesic bounds on orbifolds with diameter limits.

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

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

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

The paper finds bounds on shortest dense curves on surfaces.

Shortest non-simple closed geodesics on hyperbolic surfaces found.

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 of second eigenvalues of hyperbolic surfaces improves bounds and investigates their behavior for large genus.

EntroPath learns manifold geometry from diffusion paths.

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

Sharp bounds found on shortest geodesic on punctured spheres.

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.

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 shortest non-separating curves on non-orientable surfaces, proving NP-hardness and tractability.

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

Paper finds shortest geodesic paths on hyperbolic surfaces.

Study shortest geodesics on flat cone spheres with conical singularities.

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.

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

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

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

Study on shortest arcs on hyperbolic surfaces with boundary.

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.

The authors found geodesics, shortest arcs, cut loci, and conjugate sets for left-invariant sub-Riemannian matric on the Lie group $SL(2)$, which is right-invariant relative to the Lie subgroup $SO(2)\subset SL(2)$ (in other words, for invariant sub-Riemannian metric on weakly symmetric space $(SL(2)\times SO(2))/SO(2)…

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

The paper proves a bound on the length of the shortest geodesic flower on certain manifolds.

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…

The authors find geodesics, shortest arcs, diameter, cut locus, and conjugate sets for left-invariant sub-Riemannian metric on the Lie group SO(3), under condition that the metric is right-invariant relative to the Lie subgroup $SO(2)\subset SO(3)$.

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 …

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.

The author finds geodesics, shortest arcs, cut locus, and conjugate sets for left-invariant sub-Riemannian metric on the Lie group $SO_0(2,1)$ under the condition that the metric is right-invariant relative to the Lie subgroup $SO(2)\subset SO_0(2,1)$.

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.

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

In this paper we prove new upper bounds for the length of a shortest closed geodesic, denoted $l(M)$, on a complete, non-compact Riemannian surface $M$ of finite area $A$. We will show that $l(M) \leq 4\sqrt{2A}$ on a manifold with one end, thus improving the prior estimate of C. B. Croke, who first established that $l…

Revisits Isomap, showing it constructs Euclidean representations of geodesic structure.

Paper develops a new algorithm to find shortest paths on surfaces.

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…

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

Improved bounds on geodesic lengths in Riemannian surfaces.

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…