arXiv research

Study geodesics on spherical polyhedra, estimating their number.

3D hyperbolic spaces have endless simple paths.

All closed geodesics are simple and non-intersecting in dimensions 3 and above.

The paper classifies geodesic orbit spaces with simple isotropy groups.

Shortest non-simple closed geodesics on hyperbolic surfaces found.

Characterizes geodesics on spheres with Morse index bounds and inequalities.

Simple geodesics on spherical tetrahedra identified for specific angles.

The study finds at least two short, simple geodesic chords on a disk with convex boundary.

We give a new proof of McShane's classification of simple cuspidal geodesics, using simple equivariant methods in the hyperbolic plane.

We obtained a complete classification of simple closed geodesics on regular tetrahedra in Lobachevsky space. Also, we evaluated the number of simple closed geodesics of length not greater than $L$ and found the asymptotic of this number as $L$ goes to infinity.

Survey on geodesics on tetrahedra in curved spaces.

Geodesic loops on tetrahedra are studied in spherical and hyperbolic spaces.

In this paper, we investigate left-invariant geodesic orbit metrics on connected simple Lie groups, where the metrics are formed by the structures of generalized flag manifolds. We prove that all these left-invariant geodesic orbit metrics on simple Lie groups are naturally reductive.

On a hyperbolic Riemann surface, given two simple closed geodesics that intersect $n$ times, we address the question of a sharp lower bound $L_n$ on the length attained by the longest of the two geodesics. We show the existence of a surface $S_n$ on which there exists two simple closed geodesics of length $L_n$ interse…

New bounds on shortest geodesic loops on a sphere.

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

This paper determines which orientable hyperbolic 3-manifolds contain simple closed geodesics. The Fuchsian group corresponding to the thrice-punctured sphere generates the only example of a complete non-elementary orientable hyperbolic 3-manifold that does not contain a simple closed geodesic. We do not assume that th…

Let $M$ be a Riemannian $2$-sphere. A classical theorem of Lyusternik and Shnirelman asserts the existence of three distinct simple non-trivial periodic geodesics on $M$. In this paper we prove that there exist three simple periodic geodesics with lengths that do not exceed $20d$, where $d$ is the diameter of $M$. We a…

Study on geodesics on high genus expander surfaces, proving filling and non-simple properties.

Random simple closed curves map Teichmüller space to geodesic currents.

Decomposes Busemann spaces into simpler structures.

The goal of the article is to provide different explicit quantifications of the non density of simple closed geodesics on hyperbolic surfaces. In particular, we show that within any embedded metric disk on a surface, lies a disk of radius only depending on the topology of the surface (and the size of the first embedded…

We consider the existence of simple closed geodesics or "geodesic knots" in finite volume orientable hyperbolic 3-manifolds. Previous results show that at least one geodesic knot always exists [Bull. London Math. Soc. 31(1) (1999) 81-86], and that certain arithmetic manifolds contain infinitely many geodesic knots [J. …

Study proper sampling for X-ray transforms on simple surfaces.

It is shown that the space of null geodesics of a star-shaped causally simple subset of Minkowski space is contactomorphic to the canonical contact structure in the spherical cotangent bundle of $\mathbb{R}^n$. In the $3$-dimensional case we prove a similar result for a large class of causally simple contractible subse…

In this paper, we study homogeneous geodesics in homogeneous Finsler spaces. We first give a simple criterion that characterizes geodesic vectors. We show that the geodesics on a Lie group, relative to a bi-invariant Finsler metric, are the cosets of the one-parameter subgroups. The existence of infinitely many homogen…

The goal of the chapter is to present certain aspects of the relationship between the study of simple closed geodesics and Teichmüller spaces.

The simple length spectrum of a Riemannian manifold is the set of lengths of its simple closed geodesics. We prove a theorem claimed by Lusternik: in any Riemannian 2-sphere whose simple length spectrum consists of only one element L, any geodesic is simple closed with length L.

We prove a quantitative estimate, with a power saving error term, for the number of simple closed geodesics of length at most $L$ on a compact surface equipped with a Riemannian metric of negative curvature. The proof relies on the exponential mixing rate for the Teichmüller geodesic flow.

Random subsurfaces of hyperbolic surfaces equidistribute to ribbon graphs.

Paper proves trajectories of Chaplygin systems are reparametrized geodesics.

We describe a new approach to the study of the set of all simple geodesics on a hyperbolic punctured torus. We introduce a valuation on the first integral homology group of the torus. This valuation associates to each homology class the length of the unique simple geodesic in it. We show that this valuation extends to …

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…

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.

We define a norm on homology of punctured tori equipped with a complete hyperbolic metric of finite volume and use it to find asymptotics on the growth of the number of simple geodesics of bounded length.

We show that on any translation surface, if a regular point is contained in a simple closed geodesic, then it is contained in infinitely many simple closed geodesics, whose directions are dense in the unit circle. Moreover, the set of points that are not contained in any simple closed geodesic is finite. We also constr…

Using geodesic length functions, we define a natural family of real codimension 1 subvarieties of Teichmüller space, namely the subsets where the lengths of two distinct simple closed geodesics are of equal length. We investigate the point set topology of the union of all such hypersurfaces using elementary methods. Fi…

Find simple geodesics in hyperbolic surfaces with bounded diameter.

It is shown that the space of null geodesics of a causally simple Lorentzian manifold is Hausdorff if it admits an open conformal embedding into a globally hyperbolic spacetime. This provides an obstruction to conformal embeddings of causally simple spacetimes into globally hyperbolic ones irrespective of curvature con…

New results on geodesic flows using curve shortening flow.

Classifies geodesic orbit spaces with abelian isotropy subgroups.

We will develop simple relations between the arc-lengths of a pair of geodesics that share common end-points. The two geodesics differ only by the requirement that one is constrained to lie in a subspace of the parent manifold. We will present two applications of our results. In the first example we explore the converg…

Luo and Tan gave a new identity for hyperbolic surfaces with/without geodesic boundary in terms of dilogarithms of the lengths of simple closed geodesics on embedded three-holed spheres or one-holed tori. However, the identity was trivial for a hyperbolic one-holed torus with geodesic boundary. In this paper we adapt t…

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

Lengths of simple closed geodesics on hyperbolic surfaces in prescribed homology classes

A simple model for unbalanced optimal transport captures key features.

In this article we introduce an approach for studying the geodesic X-ray transform and related geometric inverse problems by using Carleman estimates. The main result states that on compact negatively curved manifolds (resp. nonpositively curved simple or Anosov manifolds), the geodesic vector field satisfies a Carlema…

We give existence results for simple closed curves with prescribed geodesic curvature on $S^{2}$, which correspond to periodic orbits of a charge in a magnetic field.