arXiv research

The paper classifies 10 antipodal pairings of self-dual maps.

Paper defines conditions for projective links in projective 3-space.

The paper explores symmetric representations of links and conditions for amphichirality.

Grafting a measured lamination on a hyperbolic surface defines a self-map of Teichmuller space, which is a homeomorphism by a result of Scannell and Wolf. In this paper we study the large-scale behavior of pruning, which is the inverse of grafting. Specifically, for each conformal structure $X \in \T(S)$, pruning $X$ g…

Study on embeddings and their topological properties in R^d.

Study maximal antipodal sets in exceptional symmetric spaces.

Survey on geometry and topology of maximal antipodal sets.

Characterizes higher rank model geometries using antipodal sets.

Study of Gauss maps for minimal surfaces in a specific 3D model.

The paper defines generalized s-manifolds and explores their polars and antipodal sets.

Proves conjectures about maximal antipodal sets in symmetric and generalised symmetric spaces.

Spheres can be stretched to have larger diameter than antipodal distance.

Given any finite subset X of the sphere S^n, n>1, which includes no pairs of antipodal points, we explicitly construct smoothly immersed closed orientable hypersurfaces in Euclidean space R^{n+1} whose Gauss map misses X. In particular, this answers a question of M. Gromov.

Study finds a minimum volume for vector fields on a punctured sphere.

Researchers found a maximal antipodal set of three elements in a 7x7 sphere space.

Proves Poincaré duality for Hopf algebroids with bijective antipode.

Biholomorphisms of transport twistor spaces are rigid under certain conditions.

It is known that the antipodal set of a Riemannian symmetric space of compact type $G / K$ consists of a union of $K$-orbits. We determine the dimensions of these $K$-orbits of most irreducible symmetric spaces of compact type. The symmetric spaces we are not going to deal with are those with restricted root system $\m…

The aim of this paper is a characterization of great antipodal sets of complex Grassmannian manifolds as certain designs with the smallest cardinalities.

We give an explicit classification of maximal antipodal sets in any irreducible compact symmetric space except for spin groups and half spin groups, and some quotient symmetric spaces associated to them.

The classical Lusternik-Schnirelman-Borsuk theorem states that if a d-sphere is covered by d+1 closed sets, then at least one of the sets must contain a pair of antipodal points. In this paper, we prove a combinatorial version of this theorem for hypercubes. It is not hard to show that for any cover of the facets of a …

The paper creates a deformation retraction for homeomorphisms of the projective plane.

In the first part of this paper, we consider smooth maps from a compact orientable 3-manifold without boundary to the 2-sphere. We give a geometric criterion to decide whether two given maps are homotopic, based on the sets of points where the maps are equal or antipodal. We extend this criterion to non-singular vector…

We show that the volume of any Riemannian metric on a three sphere is bounded below by the length of the shortest closed curve that links its antipodal image. In particular, the volume is bounded below by the minimum of the length of the shortest closed geodesic and the minimal distance between antipodal points.

Call a periodic map $h$ on the closed orientable surface $Σ_g$ extendable if $h$ extends to a periodic map over the pair $(S^3, Σ_g)$ for possible embeddings $e: Σ_g\to S^3$. We determine the extendabilities for all periodical maps on $Σ_2$. The results involve various orientation preserving/reversing behalves of the p…

We prove that there is a unique real tight contact structure on the 3-ball with convex boundary up to isotopy through real tight contact structures. We also give a partial classification of the real tight solid tori with the real structure being antipodal map along longitudinal and the identity along meridional directi…

Invariant structures link to algebraic curves with specific properties.

Let $M$ be an open Riemann surface and let $Λ\subset M$ be a closed discrete subset. In this paper, we prove the existence of complete conformal minimal immersions $M\to\mathbb{R}^n$, $n\ge 3$, with prescribed values on $Λ$ and whose generalized Gauss map $M\to\mathbb{CP}^{n-1}$, $n\ge 3$, avoids $n$ hyperplanes of $\m…

Minimal surfaces in S3(2) linked to vector fields on punctured sphere.

Condition for intersection of real flag manifolds in complex flag manifold.

Rational maps structure theorem with geometric decomposition and realizability proof.

Lower bounds for surface area and volume of convex hypersurfaces.

The bridge index and superbridge index of a knot are important invariants in knot theory. We define the bridge map of a knot conformation, which is closely related to these two invariants, and interpret it in terms of the tangent indicatrix of the knot conformation. Using the concepts of dual and derivative curves of s…

Study proves uniqueness of certain minimal surfaces in spherical and hyperbolic spaces.

Study finds the volume of unit vector fields on a punctured sphere and shows their images match minimally immersed Klein bottles.

We classify $R$-spaces that admit a certain natural $Γ$-symmetric structure. We further determine the maximal antipodal sets of these structures.

We characterize the three-dimensional spaces admitting at least six or at least seven equidistant points. In particular, we show the existence of $C^\infty$ norms on $\R^3$ admitting six equidistant points, which refutes a conjecture of Lawlor and Morgan (1994, Pacific J. Math \textbf{166}, 55--83), and gives the exist…

Develops obstruction theory for a specific 4-manifold index.

In these short notes we characterize the loxodromic unit vector fields on antipodally punctured Euclidean spheres as the only ones achieving a lower bound for the volume functional depending on the Poincaré indexes around their singularities.

The paper finds multiple points in maps from sphere to Euclidean space.

Geodesic lines with specific boundaries found on a special type of manifold.

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…

We give examples of Lie-Rinehart algebras whose enveloping algebra is not a full Hopf algebroid in the sense of Bohm and Szlachanyi. We construct these examples as quotients of a canonical Lie-Rinehart algebra over a Jacobi algebra which does admit an antipode.

In this paper we study singular points of the Wigner caustic and affine $λ$--equidistants of planar curves based on shapes of these curves. We generalize the Blaschke-Süss theorem on the existence of antipodal pairs of a convex curve.

In the present paper we study the structure of the cut locus of a Randers rotational 2-sphere of revolution $(M, F = α+β)$. We show that in the case when the Gaussian curvature of the Randers surface is monotone along a meridian, the cut locus of a point $q\in M$ is a point on a subarc of the opposite half bending meri…

The study restricts Anosov subgroups of Sp(2n,R) based on subset Θ.

Given a finitely generated and projective Lie-Rinehart algebra, we show that there is a continuous homomorphism of complete commutative Hopf algebroids between the completion of the finite dual of its universal enveloping Hopf algebroid and the associated convolution algebra. The topological Hopf algebroid structure of…

Study path spaces and their homology, extending loop products and coproducts.