The paper studies the shortest closed multi-geodesics on hyperbolic surfaces as their genus grows.
problem Finding the asymptotic behavior of shortest closed multi-geodesics on hyperbolic surfaces.
method Analyzing the length of shortest filling closed multi-geodesics using hyperbolic geometry and asymptotic analysis.
result The length of a shortest filling closed multi-geodesic is uniformly comparable to a specific formula involving the genus and lengths of closed geodesics.
Shortest geodesic on curved spheres is no longer than 3 times the diameter.
problem Finding the shortest closed geodesic on spheres with positive curvature.
method Proved a new isoperimetric inequality for spheres with pinched curvature, used to improve the bound on the shortest geodesic.
result The shortest closed geodesic is no longer than 3 times the diameter of the sphere.
Study shortest non-simple geodesics on 2-orbifolds, finding unique shortest curve.
problem Finding shortest non-simple closed geodesics disjoint from orbifold points.
method Fundamental domains and hyperbolic trigonometry.
result Identified and classified all figure eight geodesics on triangle group orbifolds.
The paper finds shortest geodesic bounds on orbifolds with diameter limits.
problem Bounding the shortest closed geodesic on compact orbifolds.
method Generalizing length-bounded sweepouts to orbifolds.
result Established an inequality linking shortest geodesic length to orbifold diameter.
Improved bounds on shortest geodesics with self-intersections on hyperbolic surfaces.
problem Quantifying the complexity of non-simple closed geodesics on hyperbolic surfaces.
method Analyzing the geometry of shortest figure eight curves and constructing geodesic representatives.
result Explicit upper bounds for the length of shortest geodesics with k self-intersections improved from 512 to 128. Study geodesics and shortest arcs on Lie groups with specific metrics.
problem Characterize geodesics and shortest paths on Lie groups with sub-Riemannian metrics.
method Analytical and geometric methods to find geodesics and shortest arcs.
result Found geodesics, shortest arcs, distances, and conjugate loci for specified metrics.
Study geodesics and shortest arcs on Lie groups with specific metrics.
problem Characterize geodesics and shortest arcs in sub-Riemannian metrics on Lie groups.
method Investigated left-invariant sub-Riemannian metrics on SU(1,1)imesR and SO0(2,1)imesR. result Found geodesics, shortest arcs, cut loci, and conjugate loci.
The paper finds bounds on shortest dense curves on surfaces.
problem Finding shortest dense curves on surfaces.
method Quantitative density of closed geodesics and orthogeodesics.
result Upper bounds on shortest dense curves.
Shortest non-simple closed geodesics on hyperbolic surfaces found.
problem Finding the shortest non-simple closed geodesics on hyperbolic surfaces.
method Analyzing closed geodesics with at least k self-intersections on hyperbolic surfaces.
result The shortest non-simple closed geodesics lie on an ideal pair of pants and have length $2\arccosh(2k+1)$.
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 paper improves bounds on the shortest closed geodesic length on surfaces.
problem Finding the shortest closed geodesic on surfaces of finite area.
method Proving new upper bounds for the length of the shortest closed geodesic.
result Improved bounds on the length of the shortest closed geodesic.
Study shows shortest geodesic length on certain manifolds is limited by volume, diameter, and cover elements.
problem Bounding the length of shortest closed geodesics on Riemannian manifolds with good covers.
method Generalization of previous results using diameter, volume, and cover elements to bound geodesic length.
result Length of shortest closed geodesic is bounded by a function of volume, diameter, and cover elements.
Sharp bounds found on shortest geodesic on punctured spheres.
problem Finding the shortest closed geodesic on punctured spheres.
method Sharp curvature-free upper bounds expressed in terms of area, extremal metrics described.
result Optimal bounds for spheres with up to four ends, extended to larger numbers of punctures.
New bounds on shortest geodesic loops on a sphere.
problem Finding shortest geodesic loops on a sphere.
method Analyzing geodesic loops starting and ending at a fixed point on a sphere.
result At any point on a sphere, there are at least two distinct geodesic loops whose lengths are bounded by 8d and 14d.
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 shows shortest periodic geodesic on hyperbolic orbisphere complements figure-eight knot.
problem Shortest periodic geodesic on hyperbolic orbisphere with cone points.
method Computation of linking numbers to show homeomorphism.
result Lift of shortest periodic geodesic is homeomorphic to figure-eight knot complement.
Paper finds shortest geodesic paths on hyperbolic surfaces.
problem Finding the shortest geodesic paths on hyperbolic surfaces.
method Analyzes genus g hyperbolic surfaces to find minimal length geodesics.
result Minimal geodesic length is realized by a specific polygon.
Study shortest geodesics on flat cone spheres with conical singularities.
problem Understanding the distribution of shortest geodesics on flat cone spheres.
method Proved a recurrent relation on the distribution of the length of shortest geodesics with respect to Thurston's volume form.
result Proved a recurrent relation on the distribution of the length of shortest geodesics.
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.
problem Bounding the length of shortest periodic geodesics on curved spaces.
method Analyzing the space of closed loops and their homotopy.
result The length of a shortest periodic geodesic is bounded by 8π(n−1). Study finds shortest geodesic loops on Stiefel manifold and calculates its injectivity radius.
problem Determining the shortest geodesic loops and injectivity radius on Stiefel manifold.
method Combining bounds on sectional curvature with existing metrics.
result Exact value of the injectivity radius for a wide range of metrics.
Explicit bounds found for shortest orthogeodesics and volumes of hyperbolic manifolds.
problem Finding explicit bounds for shortest orthogeodesics and volumes of hyperbolic manifolds.
method Derived explicit estimates for functions related to volumes and orthospectra, using a new approach.
result Explicit lower bound for the length of the shortest orthogeodesic in terms of volume.
Study on shortest arcs on hyperbolic surfaces with boundary.
problem Characterize and maximize the length of shortest essential arcs on hyperbolic surfaces with geodesic boundaries.
method Analyze hyperbolic surfaces with multiple boundary components, construct surfaces with large orthosystole, and compare growth rates.
result Orthosystole grows at the same rate as Bavard's upper bound as the genus increases.
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.
problem Finding bounds on the shortest geodesics in hyperbolic 3-manifolds.
method Establishing an upper bound for the length of the nth shortest closed geodesic in terms of the volume of the manifold. result An upper bound for the length of the nth shortest closed geodesic in terms of the volume of the manifold. Study intersection numbers, lengths, and shortest geodesics on hyperbolic surfaces.
problem Understanding the relationship between intersection numbers, lengths, and shortest geodesics on hyperbolic surfaces.
method Analyzing the asymptotic behavior of interaction strength I(X) as X approaches infinity in the moduli space of compact hyperbolic surfaces.
result Determined the asymptotic behavior of interaction strength I(X) in terms of the length of the shortest geodesic sys(X).
Study on shortest geodesics crossing multiple times on hyperbolic surfaces with cusps.
problem Finding shortest geodesics crossing multiple times on hyperbolic surfaces.
method Investigates closed geodesics on hyperbolic surfaces with at least one cusp, focusing on minimal length and self-intersection numbers.
result For large enough k, self-intersection numbers are exactly k for geodesics crossing at least k times. 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)⊂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.
problem Finding bounds on the length of shortest closed geodesics in hyperbolic link complements.
method Established an upper bound for the length of an nth shortest closed geodesic as a logarithmic function of the volume of the manifold.
result An upper bound of the length of an nth shortest closed geodesic is established as a logarithmic function of the volume of the manifold.
The paper proves a bound on the length of the shortest geodesic flower on certain manifolds.
problem Finding the shortest geodesic flower on a specific class of manifolds.
method Analyzing a non-compact Riemannian manifold with locally convex ends and finite volume, proving the existence of a geodesic net with constraints on its length.
result The existence of a non-trivial geodesic flower with a bounded total length on the manifold.
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)⊂SO(3).
In this paper, we show that for any closed 4-dimensional simply-connected Riemannian manifold M with Ricci curvature ∣Ric∣≤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 ε-fills the surface.
The author finds geodesics, shortest arcs, cut locus, and conjugate sets for left-invariant sub-Riemannian metric on the Lie group SO0(2,1) under the condition that the metric is right-invariant relative to the Lie subgroup SO(2)⊂SO0(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.
problem Understanding the behavior of shortest non-simple closed geodesics on hyperbolic surfaces.
method Investigation of asymptotic behavior on random hyperbolic surfaces using the Weil-Petersson measure.
result The non-simple systole behaves like log(g) as g goes to infinity.
The study of random surfaces reveals asymptotic lengths of separating geodesics.
problem Understanding geometric properties of random hyperbolic surfaces.
method Analysis of Weil-Petersson measure and asymptotic behavior of lengths.
result The shortest separating closed geodesics have lengths about 2logg. Revisits Isomap, showing it constructs Euclidean representations of geodesic structure.
problem Nonlinear dimension reduction of manifold data.
method Revisits Isomap's rationale, clarifying its approach to constructing Euclidean representations of geodesic structure.
result Convexity is not required for shortest path distances to converge to Riemannian distances.
Paper develops a new algorithm to find shortest paths on surfaces.
problem Finding shortest paths on surfaces with defined metrics.
method Uses Taylor expansion of exponential map for numerical computation.
result Developed a new algorithm to find geodesics efficiently.
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.
problem Mapping Teichmüller space to Thurston spine.
method Equivariant deformation retraction of Teichmüller space onto a cell complex.
result Thurston spine contains points corresponding to hyperbolic surfaces with shortest geodesics forming polygons.
EntroPath learns manifold geometry from diffusion paths.
problem Learning geodesic geometry from data graphs with spurious shortcuts.
method Maximum Entropy Path Ensemble Embedding (MERW) with k-step diffusion paths.
result EntroPath converges to squared geodesic distance in the short-time limit.
Improved bounds on geodesic lengths in Riemannian surfaces.
problem Finding precise lengths of geodesics in Riemannian surfaces.
method Proved curvature-free linear length bounds on geodesics.
result Length of kextth-shortest geodesic is at most 8kd. 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…
Undergrad project: Shows geodesics coincide in Heisenberg group under two metrics.
problem Identifying geodesics in Heisenberg group under two metrics.
method Examined Heisenberg group H1 with Koranyi- and Carnot-Caratheodory metrics.
result Geodesics coincide for both metrics in Heisenberg group.