arXiv research

Paper introduces spherical knot mosaics for knot and link invariants.

Approximates surfaces using Laguerre geometry with spherical faces.

We present two identities (contiguity relation and variation formula) concerning the volume of a spherically faced simplex in the Euclidean space. These identities are described in terms of Cayley-Menger determinants and their differentials involved with hypersphere arrangements. They are derived as a limit of fundamen…

Approximates smooth surfaces using Laguerre geometry meshes.

Simple geodesics on spherical tetrahedra identified for specific angles.

The problem of classifying, upto isometry (or similarity), the orientable spherical, Euclidean and hyperbolic 3-manifolds that arise by identifying the faces of a Platonic solid is formulated in the language of Coxeter groups. In the spherical and hyperbolic cases, this allows us to complete the classification begun by…

Formula for transgressions on polyhedral manifolds, linking face volumes and outer angles.

We explore some generalizations of fullerenes F_v (simple polyhedra with v vertices and only 5- and 6-gonal faces) seen as (d-1)-dimensional simple manifolds (preferably, spherical or polytopal) with only 5- and 6-gonal 2-faces. First, finite and planar (infinite) 3-fullerenes are described. Three infinite families of …

We call "flippable tilings" of a constant curvature surface a tiling by "black" and "white" faces, so that each edge is adjacent to two black and two white faces (one of each on each side), the black face is forward on the right side and backward on the left side, and it is possible to "flip" the tiling by pushing all …

Classical H.Minkowski theorems on existence and uniqueness of convex polyhedra with prescribed directions and areas of faces as well as the well-known generalization of H.Minkowski uniqueness theorem due to A.D.Alexandrov are extended to a class of nonconvex polyhedra which are called polyhedral herissons and may be de…

New Fourier features improve high-precision approximation in large-scale problems.

The paper proves a Minkowski-like theorem for tetrahedra in dS3 and AdS3.

The classical Schläfli formula, and its ``higher'' analogs given in [SS03], are relations between the variations of the volumes and ``curvatures'' of faces of different dimensions of a polyhedra (which can be Euclidean, spherical or hyperbolic) under a first-order deformation. We describe here analogs of those formulas…

We present a generalization of Minkowski's classic theorem on the reconstruction of tetrahedra from algebraic data to homogeneously curved spaces. Euclidean notions such as the normal vector to a face are replaced by Levi-Civita holonomies around each of the tetrahedron's faces. This allows the reconstruction of both s…

The paper is centered around a new proof of the infinitesimal rigidity of convex polyhedra. The proof is based on studying derivatives of the discrete Hilbert-Einstein functional on the space of "warped polyhedra" with a fixed metric on the boundary. This approach is in a sense dual to using derivatives of the volume i…

In discrete differential geometry, it is widely believed that the discrete Gaussian curvature of a polyhedral vertex star equals the algebraic area of its Gauss image. However, no complete proof has yet been described. We present an elementary proof in which we compare, for a particular normal vector, its winding numbe…

Bordifications of hyperplane arrangements yield complexes with homotopy type of wedges of spheres.

Polyhedra's structure is uniquely defined by edge lengths and dihedral angles, even nonconvex.

Study geodesics on spherical polyhedra, estimating their number.

The abstract proves spherical surface decompositions with conical singularities.

For the n-dimensional spherical pedal curve $ped_{γ,P}$ with respect to an n-dimensional spherical unit speed curve $γ$ and a given point $P \in S^n$, we define the spherical orthotomic curve of $γ$ relative to the point $P$, and classify singularities of spherical orthotomic curves.

Study spherical curves with curvature dependent on distance to a great circle.

Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.

New findings show fundamental group is not audible in spherical space forms.

Existence and uniqueness of spherical helicoidal surfaces in 3-sphere via spherical curves.

The reductivity of a spherical curve represents how reduced the spherical curve is. It is unknown if there exists a spherical curve whose reductivity is four. In this paper we give an unavoidable set for spherical curves with reductivity four by considering 4-gons.

The paper improves inequalities for nearly spherical sets using quermassintegrals.

The reductivity of a spherical curve is the minimal number of a local transformation called an inverse-half-twisted splice required to obtain a reducible spherical curve from the spherical curve. It is unknown if there exists a spherical curve whose reductivity is four. In this paper, an unavoidable set of configuratio…

We show that we can obtain a reducible spherical curve from any non-trivial spherical curve by four or less inverse-half-twisted splices, i.e., the reductivity, which represents how reduced a spherical curve is, is four or less. We also discuss unavoidable sets of tangles for spherical curves.

The paper classifies Landsberg spherically symmetric Finsler metrics in various dimensions.

For a convex body on the Euclidean unit sphere the spherical convex floating body is introduced. The asymptotic behavior of the volume difference of a spherical convex body and its spherical floating body is investigated. This gives rise to a new spherical area measure, the floating area. Remarkably, this floating area…

Study on spherical CR manifolds with non-trivial Chern classes.

PGF kernels analyze spherical data using generalized RBF kernels.

The paper examines properties of spherical Finsler metrics, proving their semi-C-reducibility and conditions for vanishing mean stretch curvature.

Classifies involutions on spherical 3-manifolds.

The paper classifies spherically symmetric Finsler metrics as Berwaldian or Riemannian.

Paper develops invariants for spherical curves using chord diagrams.

In this paper we consider the spherical slant helices in $R^3$. More- over, we show how could be obtained to a spherical slant helix and we give some spherical slant helix examples in Euclidean 3-space.

The paper characterizes spherically symmetric metrics with scalar curvature.

The study proves spherical polygon analogs of curve theorems, finding bounds on intersections and inflections.

New formula for spherical polygon area via prequantization.

Euler explored spherical geometry using trigonometric formulae and solid geometry methods.

A spherical polyhedron surface is a triangulated surface obtained by isometric gluing of spherical triangles. For instance, the boundary of a generic convex polytope in the 3-sphere is a spherical polyhedron surface. This paper investigates these surfaces from the point of view of inner angles. A rigidity result is obt…

Recently we initiated the study of spherical T-duality for spacetimes that are principal SU(2)-bundles. In this paper, we extend spherical T-duality to spacetimes that are oriented non-principal SU(2)-bundles. There are several interesting new examples in this case and a new phenomenon appearing in the non-principal ca…

Kähler-Ricci flow on spherical Fano manifolds converges to a soliton.

Spherical Plateau problem studies minimal surfaces in quotients of spheres.

The class of spherically symmetric Finsler metrics is studied and locally dually flat and projectively flat spherically symmetric Finsler metrics is classified.

In earlier papers, we introduced spherical T-duality, which relates pairs of the form $(P,H)$ consisting of an oriented $S^3$-bundle $P\rightarrow M$ and a 7-cocycle $H$ on $P$ called the 7-flux. Intuitively, the spherical T-dual is another such pair $(\hat P, \hat H)$ and spherical T-duality exchanges the 7-flux with …