arXiv research

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

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 …

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.

Study maximal antipodal sets in exceptional symmetric spaces.

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…

Survey on geometry and topology of maximal antipodal sets.

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 paper classifies 10 antipodal pairings of self-dual maps.

Characterizes higher rank model geometries using antipodal sets.

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.

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.

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.

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…

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.

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.

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

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

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

Characterizes conical angles for metrics with dihedral symmetry.

It is known that a closed polygon P is a critical point of the oriented area function if and only if P is a cyclic polygon, that is, $P$ can be inscribed in a circle. Moreover, there is a short formula for the Morse index. Going further in this direction, we extend these results to the case of open polygonal chains, or…

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…

We classify the volume preserving stable hypersurfaces in the real projective space $\mathbb{RP}^n$. As a consequence, the solutions of the isoperimetric problem are tubular neighborhoods of projective subspaces $\mathbb{RP}^k\subset \mathbb{RP}^n$ (starting with points). This confirms a conjecture of Burago and Zalgal…

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

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

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.

Study shows curvature rigidity of specific metric types.

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

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

New insights show stochastic initialization prevents token clustering in deep Transformers.

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…

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.

Study sharp upper bounds for Aharonov-Bohm eigenvalues on surfaces.

The paper proves rigidity for warped product spaces with degenerate ends.

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

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.

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

In this paper, we investigate simultaneous properties of a convex integrand $γ$ and its dual $δ$. The main results are the following three. (1) For a $C^\infty$ convex integrand $γ: S^n\to \mathbb{R}_+$, its dual convex integrand $δ: S^n\to \mathbb{R}_+$ is of class $C^\infty$ if and only if $γ$ is a strictly convex in…

The paper proves the existence of minimal surfaces avoiding specific points.

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

New theorem connects distant points and identical points on manifolds.

Biholomorphisms of transport twistor spaces are rigid under certain conditions.

Study dihedral spherical surfaces and their foliations.

By using the notion of a rigid R-matrix in a monoidal category and the Reshetikhin--Turaev functor on the category of tangles, we review the definition of the associated invariant of long knots. In the framework of the monoidal categories of relations and spans over sets, by introducing racks associated with pointed gr…