Round spheres are uniquely characterized by half-geodesics.
problem Characterizing round spheres in Riemannian geometry.
method Establishing that Riemannian spheres with specific geodesic properties are round.
result Riemannian spheres with all geodesics closed and many half-geodesics are round.
The paper extends geodesic orbit sphere classification to Finsler geometry.
problem Classifying geodesic orbit spheres in Finsler geometry.
method Generalized from Riemannian to Finsler geometry, proving constant curvature conditions.
result Geodesic orbit Finsler spheres with constant flag curvature are Randers.
Homogeneous Finsler spheres with constant curvature have specific geodesic properties.
problem Existence and properties of homogeneous Finsler spheres with constant flag curvature.
method Proofs and analysis of geodesic properties on homogeneous Finsler spheres.
result Homogeneous Finsler spheres with constant flag curvature are either Riemannian or Randers.
Proves stability of convex spheres with similar geodesic lengths.
problem Stability of convex spheres with specific geodesic properties.
method Proves C^0 Cheeger-Gromov closeness to the round sphere.
result Strictly convex 2-spheres are close to the round sphere.
Classifies totally geodesic submanifolds in Hopf-Berger spheres.
problem Classifying totally geodesic submanifolds in Hopf-Berger spheres.
method Investigates Hopf-Berger spheres, a special family of homogeneous spaces diffeomorphic to spheres constructed via Hopf fibrations.
result Discovered intriguing examples of totally geodesic submanifolds, including isometric submanifolds of real projective spaces and non-congruent submanifolds.
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.
Hamiltonian structures generalize geodesic properties on spheres.
problem Generalizing geodesic properties on spheres to Hamiltonian structures.
method Real Hamiltonian structures on RP3 and reversible Finsler 2-spheres. result All geodesics have equal lengths on certain Hamiltonian structures.
The study proves the existence of geodesics on reversible Finsler spheres.
problem Existence of closed geodesics on Finsler 2-spheres.
method Generalization of Grayson's curve shortening flow.
result Existence of three simple closed geodesics and infinitely many closed geodesics.
Geodesic graphs for special Finsler metrics on spheres are studied.
problem Characterizing geodesic orbit Finsler metrics on spheres.
method Explicit constructions and group extensions.
result Not all projective spaces admit invariant Finsler metrics.
Characterizes geodesics on spheres with Morse index bounds and inequalities.
problem Understanding geodesics on spheres using Morse theory.
method Morse-theoretic characterization and strong Morse inequalities.
result Existence of geodesics with specific Morse indices on spheres.
Geodesic nets on flat spheres are studied using Gauss-Bonnet theorem.
problem Existence and non-existence of specific geodesic nets on flat spheres.
method The theorem of Gauss-Bonnet is applied to demonstrate results.
result Existence and non-existence of geodesic nets on regular doubled polygons.
The paper finds geodesics on specific Finsler spheres with unique properties.
problem Identifying geodesics on Finsler spheres with given curvature constraints.
method Analyzes Finsler 4-spheres with specific curvature conditions to determine geodesic properties. result Proves existence of at least four prime closed geodesics under certain conditions.
For odd-dimensional spheres, there's always a second short geodesic.
problem Finding the second shortest closed geodesic on odd-dimensional spheres.
method Analyzing non-reversible Finsler metrics on spheres of odd dimension.
result There is a second closed geodesic with Morse index ≤ 4(m+2)(m-1)+2.
Characterizes curves on geodesic spheres and totally geodesic hypersurfaces in hyperbolic and spherical spaces.
problem Characterizing curves in curved spaces.
method Rotation minimizing frames and exponential maps.
result Characterizes geodesic spherical curves in hyperbolic and spherical spaces through linear equations.
Concrete example of 2-sphere with constant curvature 1 and closed geodesics.
problem Constructing a 2-sphere with constant curvature 1 and closed geodesics.
method Constructing a concrete example of a Finsler metric on the 2-sphere.
result A 2-sphere with constant Gauss curvature 1 and all geodesics closed.
The paper finds at least two elliptic closed geodesics on certain Finsler spheres.
problem Proving the existence of closed geodesics on Finsler spheres with specific curvature conditions.
method Analyzing the flag curvature and reversibility conditions to deduce the existence of geodesics.
result At least two elliptic closed geodesics exist on Finsler spheres under given curvature constraints.
Study on shortest geodesics on specific surfaces.
problem Computing shortest closed geodesics on specific surfaces.
method Computed shortest and second shortest geodesics on hyperbolic surfaces.
result Computed the number of systoles and 2-systoles on specified surfaces.
Study on sphere widths and geodesic multiplicity.
problem Understanding the width of spheres and geodesic multiplicity.
method Computed sphere widths for k=1 to 8 and used min-max critical varifolds.
result Unstable geodesics can arise with multiplicity.
The paper finds at least three non-hyperbolic closed geodesics on positively curved Finsler spheres.
problem Finding closed geodesics on Finsler spheres with specific curvature conditions.
method Analyzing Finsler n-dimensional spheres with reversibility and flag curvature constraints. result Existence of at least three non-hyperbolic closed geodesics on positively curved Finsler spheres.
Geodesic spheres in Sn+1 stable under constrained curvature flows.
problem Stability of geodesic spheres in Sn+1 under curvature flows. method Proving stability under perturbations preserving a volume type quantity.
result Geodesic spheres are stable under constrained curvature flows.
The study finds the existence of multiple closed geodesics on Finsler spheres.
problem Finding the multiplicity of closed geodesics on Finsler spheres.
method Analyzing the properties of prime closed geodesics and their irrational ellipticity.
result Existence of either exactly 2\left[\frac{n+1}{2}
ight] or infinitely many distinct closed geodesics.
A theorem proving all geodesics on a sphere are simple and of same length.
problem Characterizing Zoll Riemannian metrics on a 2-sphere.
method Analyzing the simple length spectrum of a 2-sphere.
result All geodesics on a sphere are simple and of the same length.
We found all lengths of closed sub-Riemannian paths on a sphere.
problem Finding lengths of closed sub-Riemannian paths on a sphere.
method Elementary methods avoiding explicit formulas.
result Determined all lengths of closed sub-Riemannian geodesics.
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.
Geodesic disks maximize the first non-trivial Neumann eigenvalue on spheres.
problem Maximizing the first non-trivial Neumann eigenvalue on spheres.
method Proving maximizers are geodesic disks.
result Geodesic disks maximize the first non-trivial Neumann eigenvalue.
A sphere has at least two geodesics whose product length is bounded by a constant times the area.
problem Existence of distinct geodesics on a sphere.
method Proved existence of two distinct closed geodesics with lengths satisfying a specific inequality.
result Existence of two distinct closed geodesics with lengths satisfying L1L2≤C⋅Area(S2,g). The paper studies geodesics and isoparametric functions on Finsler spheres.
problem Analyzing geodesics and isoparametric functions on Finsler spheres.
method Global expressions of geodesics and isoparametric functions derived using navigation and Cartan-Münzner polynomials.
result Construction of isoparametric families and focal submanifolds.
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.
Lower bounds on geodesic lengths for spheres with Willmore energy.
problem Finding shortest closed geodesics on spheres with Willmore energy.
method Proving a lower bound on geodesic lengths for spheres with Willmore energy below 6π.
result The energy threshold of 6π is optimal and the inequality cannot be extended to higher genus surfaces.
The study counts distinct geodesics on surfaces to characterize the round sphere.
problem Understanding the round two-dimensional sphere through its closed geodesics.
method Counting and analyzing distinct closed geodesics of bounded length.
result Characterizations of the round two-sphere based on geodesics.
Geodesic spheres in certain symmetric spaces are quantitatively stable under small perturbations.
problem Stability of geodesic spheres in symmetric spaces under perturbations.
method Quantitative stability analysis using spectral gap of the Laplacian on geodesic spheres.
result Geodesic spheres are uniformly stable with respect to small C1-volume preserving perturbations. 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.
3D spheres can't be swept by short curves, complicating geodesic length estimates.
problem Obstructing geodesic length estimates in 3D spheres.
method Constructing specific 3D spheres with controlled diameter and volume.
result Min-max methods for geodesic lengths fail for certain 3D spheres.
Proves magnetic geodesic flow on sphere is integrable with constant 2-form.
problem Proving integrability of magnetic geodesic flow on sphere.
method Analyzes magnetic geodesic flow on sphere with constant 2-form.
result Proves Liouville integrability of magnetic geodesic flow on sphere.
Proves existence of curves with constant curvature in a sphere.
problem Existence of curves with constant geodesic curvature in a Riemannian 2-sphere.
method Develops a min-max scheme for a weighted length functional.
result Proves existence for almost every prescribed curvature.
The study of geodesics on a specific sphere finds bounds on their intersections and lengths.
problem Understanding the intersections and lengths of geodesics on a triply punctured sphere.
method Analyzing the self-intersection number and combinatorial length of geodesics.
result For all closed geodesics, the difference between self-intersection number and combinatorial length is at least -1.
Sharp area bounds for 2D free boundary minimal surfaces in a sphere.
problem Sharp bounds for the area of minimal surfaces in a geodesic ball of the sphere.
method Extending earlier work by Brendle and Fraser-Schoen, applying to higher dimensions.
result New sharp bounds for area of free boundary minimal surfaces in higher dimensions.
The paper proves a minimum number of closed geodesics on positively curved Finsler spheres.
problem Proving a minimum number of closed geodesics on positively curved Finsler spheres.
method Analyzing Finsler metrics on Sn with specific curvature conditions. result There exist at least n prime closed geodesics on positively curved Finsler spheres. Totally geodesic submanifolds in spheres have restricted curvature properties.
problem Characterizing submanifolds in spheres based on curvature conditions.
method Analyzing normal curvature, scalar curvature, and second fundamental form conditions.
result Compact pseudo-umbilical submanifolds in spheres are totally geodesic under specific curvature conditions.
Study harmonic function growth on manifolds with small geodesic sphere diameters.
problem Understanding harmonic function behavior on manifolds with specific sphere diameter constraints.
method Analyze complete Riemannian manifolds with sublinear geodesic sphere extrinsic diameter.
result Derive Cheng and Yau gradient estimates for harmonic functions.
Let S be a triangulated 2-sphere with fixed triangulation T. We apply the methods of thin position from knot theory to obtain a simple version of the three geodesics theorem for the 2-sphere [5]. In general these three geodesics may be unstable, corresponding, for example, to the three equators of an ellipsoid. Using a…
We prove that a riemannian metric on the 2-sphere or the projective plane can be C2-approximated by a smooth metric whose geodesic flow has an elliptic closed geodesic.
We show that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given point determine the norm ∣∇R∣ of the covariant derivative of the Riemannian curvature tensor in that point. In particular, the spectra of small geodesic spheres in a harmonic space determi…
The paper connects geodesic nets to distance function critical points.
problem Understanding the relationship between geodesic nets and distance function critical points.
method Established a relationship between geodesic nets and critical points of the distance function.
result Bounded the number of balanced points and the length of certain minimizing geodesic nets.
Study magnetic geodesics on odd spheres, computing critical energy values.
problem Understanding magnetic geodesics on odd-dimensional spheres.
method Explicit computation and analysis of submanifolds and symmetries.
result Energy values determine magnetic geodesic connectivity on spheres.
We construct examples of hyperbolic rational homology spheres and hyperbolic knot complements in rational homology spheres containing closed embedded totally geodesic surfaces.
Approximate symmetries of geodesic equations on 2-spheres are studied. These are the symmetries of the perturbed geodesic equations which represent approximate path of a particle rather than exact path. After giving the exact symmetries of the geodesic equations, two different approaches to study the approximate symmet…
Flow preserves quermassintegrals, converging to a geodesic sphere.
problem Volume preservation issue in sphere mean curvature flow.
method Introduced a mean curvature flow with a global term to keep quermassintegrals fixed.
result Flow exists for all times and converges to a geodesic sphere.