Related papers: Simple closed geodesics on regular spherical polyh…
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…
We study spherical tetrahedra with rational dihedral angles and rational volumes. Such tetrahedra occur in the Rational Simplex Conjecture by Cheeger and Simons, and we supply vast families, discovered by computational efforts, of positive…
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…
The topological (resp. geodesic) complexity of a topological (resp. metric) space is roughly the smallest number of continuous rules required to choose paths (resp. shortest paths) between any points of the space. We prove that the geodesic…
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…
We determine the symmetrized topological complexity of the circle, using primarily just general topology.
We show that on every compact Riemannian 2-orbifold there exist infinitely many closed geodesics of positive length.
Closed geodesic lines on an ellipsoid in d-dimensional Euclidean space are considered. Explicit algebro-geometric condition for closedness of such a geodesic is given. The obtained condition is discussed in light of theta-functions theory…
We give coordinate-minimal geometric realizations in general position of all 865 vertex-minimal triangulations of the orientable surface of genus 2 in the 4x4x4-cube.
For each odd $n \geq 3$, we construct a closed convex hypersurface of $\mathbb{R}^{n+1}$ that contains a non-degenerate closed geodesic with Morse index zero. A classical theorem of J. L. Synge would forbid such constructions for even $n$,…
A problem that is simple to state in the context of spherical geometry, and that seems rather interesting, appears to have been unexamined to date in the mathematical literature. The problem can also be recast as a problem in the real…
We prove that there are thirteen Archimedean/semiregular polyhedra by using Euler's polyhedral formula.
For a finite planar graph, it associates with some metric spaces, called (regular) spherical polyhedral surfaces, by replacing faces with regular spherical polygons in the unit sphere and gluing them edge-to-edge. We consider the class of…
We prove that any simple polytope (and some non-simple polytopes) in $\mathbb R^3$ admits an inscribed regular octahedron.
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.
We give in this paper bounds for the Morse indices of a large class of simple geodesics on a surface with a generic metric. To our knowledge these bounds are the first that use only the generic hypothesis on the metric.
We introduce a novel representation and optimization framework for discrete geodesics on triangle meshes that reduces artifacts of linear methods on uneven and coarse discretizations. Our method computes squared geodesic distances from…
We give a simple proof based on symmetries that there are no geodesics from a vertex to itself in the cube, tetrahedron, octahedron, and icosahedron.
This paper deals with triangulations of the 2-torus with the vertex labeled general octahedral graph $O_4$ which is isomorphic to the complete four-partite graph $K_{2,2,2,2}$; it is known that there exist precisely twelve such…