A half-geodesic is a closed geodesic realizing the distance between any pair of its points. All geodesics in a round sphere are half-geodesics. Conversely, this note establishes that Riemannian spheres with all geodesics closed and sufficiently many half-geodesics are round.
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48 results for “geodesic spheres”
Stability of Convex Spheresmath.DG
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.
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.
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.
The paper finds geodesics on specific Finsler spheres with unique properties.
problem Identifying geodesics on Finsler spheres with given curvature constraints.
method Analyzes Finsler -spheres with specific curvature conditions to determine geodesic properties.
result Proves existence of at least four prime closed geodesics under certain conditions.
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.
We prove that a homogeneous Finsler sphere with constant flag curvature and a prime closed geodesic of length must be Riemannian. This observation provides the evidence for the non-existence of homogeneous Bryant spheres. It also helps us propose an alternative approach proving that a geodesic orbit Fin…
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.
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.
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 .
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.
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.
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.
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 -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.
In this paper, we generalize the classification of geodesic orbit spheres from Riemannian geometry to Finsler geometry. Then we further prove if a geodesic orbit Finsler sphere has constant flag curvature, it must be Randers. It provides an alternative proof for the classification of invariant Finsler metrics with $K\e…
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.
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 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.
We extend two celebrated theorems on closed geodesics of Riemannian 2-spheres to the larger class of reversible Finsler 2-spheres: Lusternik-Schnirelmann's theorem asserting the existence of three simple closed geodesics, and Bangert-Franks-Hingston's theorem asserting the existence of infinitely many closed geodesics.…
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 with specific curvature conditions.
result There exist at least prime closed geodesics on positively curved Finsler spheres.
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.
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 show that in any harmonic space, the eigenvalue spectra of the Laplace operator on small geodesic spheres around a given point determine the norm 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…
We construct examples of hyperbolic rational homology spheres and hyperbolic knot complements in rational homology spheres containing closed embedded totally geodesic surfaces.
The Hopf-Rinow theorem and the Mañé critical value for magnetic geodesics on odd-dimensional spheresmath.SG
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.
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…
Study shows magnetic trajectories in Berger spheres are homogeneous.
problem Homogeneity of contact magnetic trajectories in Berger spheres.
method Proved every contact magnetic trajectory is a product of a homogeneous geodesic and a charged Reeb flow.
result Contact magnetic trajectories in Berger spheres are homogeneous.
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.
We prove that for every $\Q$-homological Finsler 3-sphere with a bumpy and irreversible metric , either there exist two non-hyperbolic prime closed geodesics, or there exist at least three prime closed geodesics.
Introduces mobility algebra for modeling geodesics on n-spheres.
problem Modeling geodesics on n-spheres using algebraic structures.
method Introduces mobility algebra and mobility spaces, showing connections to modules and affine spaces.
result Shows geodesics on n-spheres as mobility spaces over unit interval mobility algebra.
A Finsler space is said to be geodesically reversible if each oriented geodesic can be reparametrized as a geodesic with the reverse orientation. A reversible Finsler space is geodesically reversible, but the converse need not be true. In this note, building on recent work of LeBrun and Mason, it is shown that a geodes…
Study proves existence of closed geodesics on spheres and projective spaces.
problem Proving the existence of closed geodesics on Finsler metrics.
method Topological methods and Lusternik-Schnirelmann-type approach, using spherical complexities.
result Existence of multiple closed geodesics and upper bounds on their lengths.
Totally geodesic hypersurfaces in a sphere have small total curvature.
problem Characterizing hypersurfaces with constant scalar curvature in a sphere.
method Analyzing the total curvature of locally conformally flat hypersurfaces.
result Hypersurfaces with small total curvature are totally geodesic.
Reconstruction along a geodesic from sphere data in Finsler geometry and anisotropic elasticitymath.DG
Reconstructing Finsler manifolds from sphere data.
problem Recovering a Finsler manifold from sphere data.
method Solving the geometrical inverse problem locally along geodesics.
result Local reconstruction of Finsler manifolds.
New geometric invariant from min-max width of spheres on Riemannian 2-spheres.
problem Understanding the min-max width of spheres associated to distance functions.
method Application of min-max methods to pairs of points on Riemannian 2-spheres.
result The min-max width does not always equal half the length of a simple closed geodesic.
Study confirms a 2-sphere metric with three geodesics of minimal length.
problem Understanding the systolic, width, and Gromov-Guth metrics on a 2-sphere.
method Classical min-max and hyperbolic geometry tools.
result Figure-eight geodesics achieve the systolic, width, and Gromov-Guth metrics on a 2-sphere.
The consideration of the so-called rotation minimizing frames allows for a simple and elegant characterization of plane and spherical curves in Euclidean space via a linear equation relating the coefficients that dictate the frame motion. In this work, we extend these investigations to characterize curves that lie on a…
Study extends geodesic curvature formula to higher dimensions.
problem Extending curvature formula to higher-dimensional spheres.
method Using new integral-geometric formulas for Euclidean and geodesic total curvature.
result Explicit formula for geodesic total curvature on higher-dimensional spheres.
New method to bound Laplacian eigenvalues of geodesic balls.
problem Computing upper bounds for the first eigenvalue of Laplacian on geodesic balls.
method Transforming metric tensor into rotationally symmetric form preserving geodesic sphere areas.
result Upper bound for Laplacian eigenvalues is sharp and computable using geodesic sphere areas.
Research extends geodesic length function study to three holed sphere.
problem Geodesic length function on orbifolds.
method Extending previous work on punctured torus to three holed sphere and related orbifolds.
result Extension to three holed sphere and related orbifolds.
Existence of minimal annuli in 3-sphere with boundary on geodesic spheres.
problem Existence of free boundary minimal annuli in 3-sphere.
method One-parameter family of complete minimal immersions of R × S^1 into S^3, analysis of Otsuki tori.
result Existence of embedded free boundary minimal annuli contained in geodesic balls.
Study distance maps on spaces with curvature bound, proving regularity and sphere theorem.
problem Regularity of distance maps on geodesically complete spaces with curvature bound above.
method Define and prove regularity of distance maps as Hurewicz fibrations.
result Sphere theorem for geodesically complete CAT(1) spaces.
We prove the absence of a universal diameter bound on lengths of curves in a sweep-out of a Riemannian 2-sphere. If such bound existed it would yield a simple proof of existence of short geodesic segments and closed geodesics on a sphere of small diameter.