Study maximal antipodal sets in exceptional symmetric spaces.
problem Classify maximal antipodal sets in exceptional symmetric spaces.
method Combining existing literature and new results, classify maximal antipodal sets.
result Complete classification of maximal antipodal sets in all exceptional compact symmetric spaces.
Proves conjectures about maximal antipodal sets in symmetric and generalised symmetric spaces.
problem Cohomological descriptions of maximal antipodal sets in symmetric spaces.
method Equivariant cohomology theory.
result Proves several long-standing conjectures by Chen--Nagano and extends them to generalised symmetric spaces.
Characterizes higher rank model geometries using antipodal sets.
problem Identifying higher rank model geometries among Hadamard spaces.
method Using antipodal sets at infinity to characterize model geometries.
result Characterizes Riemannian symmetric spaces, Euclidean buildings, and products as higher rank model geometries.
The paper defines generalized s-manifolds and explores their polars and antipodal sets.
problem Understanding polars and antipodal sets in generalized s-manifolds.
method Introduced generalized s-manifolds and provided a method to construct them. Studied polars and antipodal sets.
result Extended results on compact symmetric spaces to generalized s-manifolds.
Researchers found a maximal antipodal set of three elements in a 7x7 sphere space.
problem Determining the maximal antipodal set in the outer 3-symmetric space S7imesS7. method Investigated the polar and maximal antipodal set P for the given 3-symmetric space S7imesS7. result The maximal antipodal set P has three elements. Classifies R-spaces with a specific symmetric structure.
problem Classifying R-spaces with a natural Γ-symmetric structure.
method Classification and determination of maximal antipodal sets.
result Classification of R-spaces with a natural Γ-symmetric structure.
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.
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 paper explores symmetric representations of links and conditions for amphichirality.
problem Investigating symmetric representations of links and conditions for amphichirality.
method Using antipodally self-dual and antipodally symmetric maps, the authors provide sufficient combinatorial conditions for amphichirality.
result A link is amphichiral if its self-dual pairing is not one of 6 specific ones.
Paper defines conditions for projective links in projective 3-space.
problem Characterizing links in projective 3-space.
method Combinatorial conditions and antipodal symmetry.
result Easy condition to prevent alternating projective links.
Condition for intersection of real flag manifolds in complex flag manifold.
problem Intersection conditions of real flag manifolds in a complex flag manifold.
method Condition given in terms of symmetric triad, antipodal intersection proven.
result Intersection of real flag manifolds is antipodal.
Survey on geometry and topology of maximal antipodal sets.
problem Maximal antipodal sets on Riemannian manifolds.
method Comprehensive survey of existing research.
result Relation to various mathematical areas.
The paper classifies 10 antipodal pairings of self-dual maps.
problem Understanding the antipodal pairings of strongly involutive polyhedra.
method Classification of self-dual pairings and construction of polyhedra.
result Determination of 10 antipodal pairings among 24 self-dual pairings.
Spheres can be stretched to have larger diameter than antipodal distance.
problem Finding the maximum diameter of spheres relative to antipodal points.
method Deformation of spheres in dimensions ≥3.
result Spheres can be deformed to have diameter larger than antipodal distance.
Study finds a minimum volume for vector fields on a punctured sphere.
problem Finding the minimum volume of unit vector fields on a punctured sphere.
method Analyzes the volume of vector fields tangent to an antipodally punctured unit 2-sphere.
result Provides a lower bound for the volume of unit vector fields.
Proves Poincaré duality for Hopf algebroids with bijective antipode.
problem Proving Poincaré duality for Hopf algebroids.
method Using twisted Poincaré duality and bijective antipode properties.
result Recovering and extending known Poincaré dualities for Hopf algebroids.
We create a minimal triangulation of 5D real projective space.
problem Tackling the minimal triangulation of 5D real projective space.
method Constructing a 6-dimensional polytope with a highly symmetric automorphism group.
result Our construction uses the fewest number of vertices (24) for a triangulation of 5D real projective space.
The aim of this paper is a characterization of great antipodal sets of complex Grassmannian manifolds as certain designs with the smallest cardinalities.
We prove a uniqueness theorem for immersed spheres of prescribed (non-constant) mean curvature in homogeneous three-manifolds. In particular, this uniqueness theorem proves a conjecture by A.D. Alexandrov about immersed spheres of prescribed Weingarten curvature in R3 for the special but important case of prescribed me…
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 …
A left-invariant sub-Riemannian metric d on the shortened Lorentz group SO0(2,1) under the condition that d is right-invariant relative to the orthogonal Lie subgroup 1⊗SO(2) is studied. The distance between arbitrary two elements, the cut locus (as the union of the subgroup 1⊗SO(2) with the an…
Polyhedral surfaces can be broken down into parallelograms.
problem Decomposing polyhedral surfaces into simpler shapes.
method Analyzing moduli spaces and using geometric properties.
result Polyhedral surfaces with 8 vertices can be decomposed into at most 20 parallelograms.
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.
The total diameter of a closed planar curve C⊂R2 is the integral of its antipodal chord lengths. We show that this quantity is bounded below by twice the area of C. Furthermore, when C is convex or centrally symmetric, the lower bound is twice as large. Both inequalities are sharp and the equality holds i…
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…
Minimal surfaces in S3(2) linked to vector fields on punctured sphere.
problem Connecting minimal surfaces in S3(2) to vector fields on a punctured sphere.
method Established a correspondence between minimal surfaces and area-minimizing vector fields.
result Stability relation for Lawson cylinders in S3(2).
Study on Borel Anosov subgroups in SL(d,R) for d≠5,8k±1.
problem Characterizing Borel Anosov subgroups in SL(d,R).
method Analysis of antipodal subsets and quasi-isometric embeddings.
result Borel Anosov subgroups are virtually free or hyperbolic surface groups.
Study finds the volume of unit vector fields on a punctured sphere and shows their images match minimally immersed Klein bottles.
problem Finding the volume of unit vector fields on a punctured sphere.
method Analyzes the volume of unit vector fields on an antipodally punctured unit 2-sphere and shows their images coincide with minimally immersed Klein bottles.
result The images of minimizing vector fields on the punctured sphere match those of minimally immersed Klein bottles.
Springer varieties are studied because their cohomology carries a natural action of the symmetric group Sn and their top-dimensional cohomology is irreducible. In his work on tangle invariants, Khovanov constructed a family of Springer varieties Xn as subvarieties of the product of spheres (S2)n. We show that…
Study on embeddings and their topological properties in R^d.
problem Topology of embeddings and their bounds in R^d.
method Combinatorial formula for upper bounds of embeddings into R^d.
result Simple combinatorial formula for upper bounds of embeddings into R^d.
We characterize the three-dimensional spaces admitting at least six or at least seven equidistant points. In particular, we show the existence of C∞ norms on R3 admitting six equidistant points, which refutes a conjecture of Lawlor and Morgan (1994, Pacific J. Math \textbf{166}, 55--83), and gives the exist…
The study examines special domains in S^2 supporting specific solutions to a PDE.
problem Identifying special domains in S^2 supporting positive solutions to a PDE.
method Extends moving plane method and Alexandrov reflection method to prove symmetry.
result Domains must be rotationally symmetric under specific conditions.
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.
problem Deformation retraction of homeomorphisms of the projective plane.
method Equivariant strong deformation retraction from homeomorphism group to special orthogonal group.
result Induces a SO(3)-equivariant strong deformation retraction from projective plane homeomorphisms to SO(3).
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.
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.
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∈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.
problem Characterizing Anosov subgroups of Sp(2n,R) based on subset Θ.
method Analyzing the structure of Anosov subgroups in terms of subset Θ.
result Anosov subgroups of Sp(2n,R) are virtually free or surface groups if Θ contains an odd integer, otherwise they are not.
Biholomorphisms of transport twistor spaces are rigid under certain conditions.
problem Rigidity of biholomorphisms in transport twistor spaces.
method Proof of rigidity for biholomorphisms between transport twistor spaces of simple or Anosov surfaces.
result Biholomorphisms are rigid, up to constant rescaling and the antipodal map, being lifts of orientation-preserving isometries.
Characterizes loxodromic unit vector fields on punctured spheres.
problem Finding vector fields with a lower bound on volume functional.
method Characterization based on Poincaré indexes.
result Only loxodromic unit vector fields achieve the lower bound.
Study path spaces and their homology, extending loop products and coproducts.
problem Understanding the homology of path spaces in closed manifolds.
method Morse-Bott theory and homology operations.
result Complete computation of extended loop product and coproduct on spheres.
We provide explicit, simple, geometric formulas for free involutions rho of Euclidean spheres that are not conjugate to the antipodal involution. Therefore the quotient S^n/rho is a manifold that is homotopically equivalent but not diffeomorphic to RP^n. We use these formulas for constructing explicit non-trivial eleme…
Characterizes conical angles for metrics with dihedral symmetry.
problem Understanding metrics with specific symmetry properties.
method Using recent results on local invariants of quadratic differentials.
result Complete characterization of conical angles for dihedral spherical metrics.
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…
Study of Gauss maps for minimal surfaces in a specific 3D model.
problem Characterizing minimal surfaces in a non-standard 3D space.
method Defining and analyzing Gauss maps for surfaces in S2imesR, proving properties of these maps. result Minimal surfaces with the same non-constant Gauss map are related by specific isometries.
The study extends inscription problems to non-Euclidean geometries.
problem Generalizing inscription problems to non-Euclidean geometries.
method Symplectic and Riemannian geometry techniques.
result Proved generalized inscription theorems for hyperbolic and spherical surfaces.
Invariant structures link to algebraic curves with specific properties.
problem Linking invariant hypercomplex structures to algebraic curves.
method Mapping invariant structures to algebraic curves with specific properties.
result Invariant hypercomplex structures correspond to algebraic curves with a flat projection and antiholomorphic involution.
The paper refines Steinerberger curvature for block graphs and bridges.
problem Understanding curvature in graph theory.
method Formulas and relations for curvature in block graphs and graph bridges.
result Self-centered Bonnet-Myers sharp graphs are antipodal.