Related papers: Simple Closed Quasigeodesics on Tetrahedra
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…
The space of shapes of a polyhedron with given total angles less than 2\pi at each of its n vertices has a Kaehler metric, locally isometric to complex hyperbolic space CH^{n-3}. The metric is not complete: collisions between vertices take…
A cubic polyhedron is a polyhedral surface whose edges are exactly all the edges of the cubic lattice. Every such polyhedron is a discrete minimal surface, and it appears that many (but not all) of them can be relaxed to smooth minimal…
We determine (non-necessarily convex) polyhedra having simple dense geodesics.
We show that every closed Lorentzian surface contains at least two closed geodesics. Explicit examples show the optimality of this claim. Refining this result we relate the least number of closed geodesics to the causal structure of the…
It is known that we can always 3-triangulate (i.e. divide into tetrahedra) convex polyhedra but not always non-convex ones. Polyhedra topologically equivalent to sphere with $p$ handles, shortly $p$-toroids, could not be convex. So, it is…
We prove that every homogeneous convex polyhedron with only one unstable equilibrium (known as a mono-unstable convex polyhedron) has at least $7$ vertices. Although it has been long known that no mono-unstable tetrahedra exist, and…
We show that cusped finite-volume hyperbolic 3-manifolds contain infinitely many simple closed geodesics.
Toeplitz's Square Peg Problem asks whether every continuous simple closed curve in the plane contains the four vertices of a square. It has been proved for various classes of sufficiently smooth curves, some of which are dense, none of…
We analyze polyhedra composed of hexagons and triangles with three faces around each vertex, and their 3-regular planar graphs of edges and vertices, which we call "trihexes". Trihexes are analogous to fullerenes, which are 3-regular planar…
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…
We prove that there are thirteen Archimedean/semiregular polyhedra by using Euler's polyhedral formula.
We review the theory of intrinsic geometry of convex surfaces in the Euclidean space and prove the following theorem: if the surface of a convex body K contains arbitrary long closed simple geodesics, then K is an isosceles tetrahedron.
We prove Calegari's conjecture that every quasigeodesic flow on a closed hyperbolic 3-manifold has closed orbits.
In this work we solve a couple of well known open problems related to the quasihyperbolic metric. In the case of planar domains, our first main result states that quasihyperbolic geodesics are unique in simply connected domains. As the…
Given a combinatorial description $C$ of a polyhedron having $E$ edges, the space of dihedral angles of all compact hyperbolic polyhedra that realize $C$ is generally not a convex subset of $\mathbb{R}^E$ \cite{DIAZ}. If $C$ has five or…
We prove that, given a polyhedron $\mathcal P$ in $\mathbb{R}^3$, every point in $\mathbb R^3$ that does not see any vertex of $\mathcal P$ must see eight or more edges of $\mathcal P$, and this bound is tight. More generally, this remains…
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],…
We prove the existence of multiple closed geodesics on non-compact cylindrica manifolds.
In this paper, we prove the existence of at least two distinct closed geodesics on every compact simply connected irreversible or reversible Finsler (including Riemannian) manifold of dimension not less than 2.