The Pachner graph of 2-spheres is studied, focusing on subgraphs of flag and stacked 2-spheres.
problem Characterize subgraphs of the Pachner graph of 2-spheres.
method Analyzes various induced subgraphs of the Pachner graph of n-vertex triangulated 2-spheres, proving connectivity and providing bounds on the number of connected components. result The subgraph of n-vertex flag 2-spheres is connected, while the subgraph of n-vertex stacked 2-spheres has at least as many connected components as trees with specific properties. Study on flag manifolds using Hopf fibration to find geometrically special 2-spheres.
problem Computing and characterizing 2-spheres in flag manifolds.
method Hopf fibration, invariant geometry, Weyl group action.
result Generators of second homotopy group have invariant geometry and are classified.
A Finsler space is said to be geodesically reversible if each oriented geodesic can be reparametrized as a geodesic with the reverse orientation. A reversible Finsler space is geodesically reversible, but the converse need not be true. In this note, building on recent work of LeBrun and Mason, it is shown that a geodes…
This article is an exposition of four loosely related remarks on the geometry of Finsler manifolds with constant positive flag curvature. <p> The first remark is that there is a canonical Kahler structure on the space of geodesics of such a manifold. <p> The second remark is that there is a natural way to construct a (…
In 1987, Kalai proved that stacked spheres of dimension d≥3 are characterised by the fact that they attain equality in Barnette's celebrated Lower Bound Theorem. This result does not extend to dimension d=2. In this article, we give a characterisation of stacked 2-spheres using what we call the {\em separatio…
We give a lower bound for the length of a non-trivial geodesic loop on a simply-connected and compact manifold of even dimension with a non-reversible Finsler metric of positive flag curvature. Harris and Paternain use this estimate in their recent paper [HP] to give a geometric characterization of dynamically convex F…
We consider the geodesic flow of reversible Finsler metrics on the 2-sphere and the 2-torus, whose geodesic flow has vanishing topological entropy. Following a construction of A. Katok, we discuss examples of Finsler metrics on both surfaces, which have large ergodic components for the geodesic flow in the unit tangent…
The study classifies discrete pseudomanifolds with up to 2d+7 vertices.
problem Understanding discrete pseudomanifolds with a small number of vertices.
method Proved existence of at least 2(d+1) vertices, classified up to 2d+6 vertices, established equivalence with edge graphs of flag normal pseudomanifolds.
result Every flag normal d-pseudomanifold with at most 2d+7 vertices is either a simplicial d-sphere or a flag triangulation of the (d-2)-fold suspension of RP^2.
Solves a triangulation problem by showing minimum tetrahedra equals minimum integral 3-chain.
problem Finding the minimum number of tetrahedra to extend a triangulation of a 2-sphere to a 3-ball.
method Relates the minimum number of tetrahedra to the minimum integral 3-chain norm, proving them equal and showing how to achieve the minimum.
result The minimum number of tetrahedra needed to extend a triangulation of a 2-sphere to a 3-ball equals the minimum integral 3-chain norm.
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.
Unique simple spines of homotopy 2-spheres are shown to be ambiently isotopic.
problem Characterizing and comparing simple spines of homotopy 2-spheres.
method Proving ambient isotopy of simple spines representing the same homology class.
result Simple spines of homotopy 2-spheres are unique.
The 3-sphere has either 2 minimal 2-spheres or an optimal foliation by 2-spheres.
problem Proving existence of minimal 2-spheres or optimal foliations in arbitrary Riemannian 3-spheres.
method Analyzing the properties of arbitrary Riemannian metrics on 3-spheres.
result The existence of at least two minimal 2-spheres or an optimal foliation in 3-spheres with arbitrary metrics.
Concrete example of 2-sphere with constant curvature 1 and closed geodesics.
problem Constructing a 2-sphere with constant curvature 1 and closed geodesics.
method Constructing a concrete example of a Finsler metric on the 2-sphere.
result A 2-sphere with constant Gauss curvature 1 and all geodesics closed.
Cominuscule subvarieties found in flag varieties.
problem Identifying special subvarieties in flag varieties.
method Using Dynkin diagrams to compute subvariety structure.
result Every flag variety has a cominuscule subvariety.
The paper studies minimal 2-spheres of constant curvature in complex hyperquadrics using matrix theory.
problem Exploring minimal 2-spheres of constant curvature in complex hyperquadrics.
method Using singular-value decomposition of complex matrices to study the moduli space of noncongruent spheres.
result Uniqueness proven for totally real constantly curved 2-spheres.
Investigates secant dimensions and identifiability in flag varieties.
problem Secant dimensions and identifiability of flag varieties.
method Numerical conditions ensuring secant varieties have expected dimension and points are identifiable.
result Secant varieties of flag varieties have expected dimension and points are identifiable under certain conditions.
Study of weighted nonlinear flags in symplectic geometry.
problem Understanding the geometry of weighted nonlinear flags.
method Generalizing weighted nonlinear Grassmannians to Frechet manifolds and using them to describe coadjoint orbits.
result Description of coadjoint orbits of Hamiltonian diffeomorphisms using weighted isotropic nonlinear flags.
New algorithm computes flag mean and median on flag manifolds.
problem Computing first order flag statistics on flag manifolds.
method Transformed problem to Stiefel manifold for optimization.
result Proved convergence and effectiveness of the flag-mean computation.
Complete obstruction found for 2-spheres in 5-manifolds to be isotopic.
problem Obstructing homotopic embeddings of 2-spheres into 5-manifolds from being isotopic.
method Using a level preserving Whitney move in codimension 3 to eliminate double points, and classical methods.
result New results for simply-connected 5-manifolds and 2-spheres with algebraic dual 3-spheres.
We study metric spaces homeomorphic to the 2-sphere, and find conditions under which they are quasisymmetrically homeomorphic to the standard 2-sphere. As an application of our main theorem we show that an Ahlfors 2-regular, linearly locally contractible metric 2-sphere is quasisymmetrically homeomorphic to the standar…
The paper studies Finsler manifolds with a new curvature concept.
problem Understanding Finsler manifolds with positive weighted flag curvature.
method Introducing a new curvature concept based on the flag curvature and a non-Riemannian quantity, T-curvature.
result Positive weighted flag curvature implies the manifold is diffeomorphic to Euclidean space.
Study nonlinear flags as coadjoint orbits of Hamiltonian diffeomorphisms.
problem Geometry of nonlinear flags and their coadjoint orbits.
method Generalization of nonlinear Grassmannians to Frechet manifolds.
result Description of symplectic nonlinear flags as coadjoint orbits.
Motivated by the rich theory of harmonic maps from a 2-sphere, we study biharmonic maps from a 2-sphere in this paper. We first derive biharmonic equation for rotationally symmetric maps between rotationally symmetric 2-manifolds. We then apply the equation to obtain a classification of biharmonic maps in a family of r…
The paper classifies Randers metrics with scalar flag curvature.
problem Classifying Finsler metrics of scalar flag curvature.
method Investigating Randers metrics under the condition that β is a Killing 1-form.
result Obtained necessary conditions for Randers metrics to be of scalar flag curvature.
Minimal dimensions found for flag manifolds embeddings.
problem Finding the smallest dimensions for flag manifolds embeddings.
method Equivariant embeddings of orthogonal and unitary groups acting on real and complex flag manifolds.
result Minimal dimensions achieved at isospectral models.
Study equigeodesics on G2-type flag manifolds, splitting tangent spaces.
problem Characterize geodesics in G2-type flag manifolds. method Analyze flag manifolds with G2-type t-roots, split tangent spaces, and classify equigeodesics. result Characterized structural equigeodesic vectors in flag manifolds.
Extends residue theory to flags of holomorphic distributions.
problem Calculating the residue class of flags of holomorphic distributions.
method Developed an effective method to calculate the class in certain cases.
result Established a relation between degrees, tangency order, Euler characteristic, and curve degree.
The paper introduces geometric surgeries for flag structures and provides examples of uniformizable and non-uniformizable types.
problem Understanding and classifying flag structures of different types.
method Introducing geometric surgeries for flag structures and applying them to examples.
result Examples of both uniformizable and non-uniformizable flag structures were provided.
Researchers prove formulas for flag area measures, extending previous work.
problem Proving additive kinematic formulas for flag area measures.
method Introducing an algebraic framework to compute these formulas explicitly.
result Existence and explicit computation of additive kinematic formulas for flag area measures.
New recursive relations for symmetries of special 2-flags discovered.
problem Local finite classification and continuous numerical modulus of special 2-flags.
method Recursive relations derived for all infinitesimal symmetries of special 2-flags.
result Existence of a continuous numerical modulus of special 2-flags in length seven.
Study finds numerical moduli in special 2-flags of length 5.
problem Identifying numerical moduli in local classifications of special multi-flags.
method Analyzing distributions generating special multi-flags, focusing on lengths up to 5.
result Three numerical moduli appear in special 2-flags of length 5.
Study finds conditions for Kähler-Einstein metrics on flag manifolds.
problem Characterizing Kähler-Einstein metrics on flag manifolds.
method Using Lie theoretic data, establish a sufficient and necessary condition for λ1-extremality. result Identifies criteria for a metric to be a critical point of the first eigenvalue functional.
The degree of certain holomorphic 2-spheres is bounded.
problem Understanding the degree of holomorphic 2-spheres in complex Grassmannians.
method Analyzing the degree of linearly full constantly curved holomorphic 2-spheres in G(2,n+2;C).
result The degree is bounded between n and 2n for specific n values.
The paper calculates expected distances on partially oriented flag manifolds.
problem Understanding distances on partially oriented flag manifolds.
method Computing expected distances on low-dimensional examples.
result Computed expected distances on partially oriented flag manifolds.
Study hyperbolic 2-spheres with cone points, describing spaces for n=3.
problem Characterize the space of hyperbolic 2-spheres with cone points.
method Analyzing the space C(a0,a1,…,an) for n=3 and n=4. result Detailed description of spaces for n=3 and examples for n=4. The paper examines the geometry of specific submanifolds in flag manifolds.
problem Understanding the geometry of invariant almost semi Kähler submanifolds.
method Analyzing homogeneous spaces as almost Hermitian submanifolds of flag manifolds.
result Minimal and totally geodesic properties of certain submanifolds.
We study metrics on conic 2-spheres when no Einstein metrics exist. In particular, when the curvature of a conic metric is positive, we obtain the best curvature pinching constant. We also show that when this best pinching constant is approached, the conic 2-sphere has an explicit Gromov-Hausdorff limit. This is a gene…
Conditions found for flag manifolds to use Bochner coordinates.
problem Finding conditions for Bochner coordinates on flag manifolds.
method Analyzing complex coordinates on flag manifolds of classical groups.
result Necessary and sufficient conditions for Bochner coordinates on flag manifolds.
Optimizes over flag manifolds for numerical PDE and statistics.
problem Optimizing over flag manifolds for numerical PDE and statistics.
method Develops tools for Riemannian optimization on flag manifolds, deriving analytic expressions and parameterizations.
result Closed-form analytic expressions and parameterizations for various geometric objects on flag manifolds.
New flag area measures constructed via integration over differential forms.
problem Constructing new flag area measures in high-dimensional spaces.
method Using local parallel sets and integration over the normal cycle of differential forms.
result Flag area measures span the space of all smooth SO(n)-covariant flag area measures.
The paper classifies complex Dirac structures on flag manifolds.
problem Classifying invariant complex Dirac structures on flag manifolds.
method Described using roots of the Lie algebra and classified under B-transformations. result All invariant complex Dirac structures with constant real index on a maximal flag manifold are described.
The paper bounds entries of γ-vectors for flag homology spheres.
problem Bounding entries of γ-vectors for flag homology spheres.
method Structural and enumerative results for flag homology spheres.
result Supports a conjecture and characterizes structures of flag homology spheres.
Berwald spaces with non-zero flag curvature are Riemannian.
problem Understanding the conditions under which Berwald spaces become Riemannian.
method Analyzing the flag curvature of Berwald spaces and proving their rigidity under certain conditions.
result Berwald spaces with non-zero flag curvature are Riemannian.
Study on invariant almost complex structures on real flag manifolds.
problem Existence of invariant almost complex structures on real flag manifolds.
method Analysis of real flag manifolds associated to split real forms of complex simple Lie algebras.
result Some real flag manifolds do not admit invariant almost complex structures.
A theorem on Finsler metrics with special curvature properties.
problem Characterizing Finsler metrics with sectional flag curvature.
method Analyzing the relationship between flag curvature and sectional curvature.
result Locally, Finsler metrics are either Riemannian or have isotropic flag curvature.
In the recent years, a number of issues concerning distributions generating 1- flags (called also Goursat flags) has been analyzed. Presently similar questions are discussed as regards distributions generating multi-flags. (In fact, only so-called special multi-flags, to avoid functional moduli.) In particular and fore…
The aim of this paper is to show that any stable complete Riemannian flag on a compact and connected manifold is conjugated to a flag of homogenus foliations (see Definitions). Also, we give a characterization of Riemannian flags that homogenus. This result is a step toward the classification of Riemannian flags.
Classifies minimal immersions from S2 into specific flag manifolds.
problem Classifying minimal immersions from S2 into specific flag manifolds. method Classification based on constant curvature and low-dimensional flag manifolds.
result Primitive minimal immersions of constant curvature from S2 into F2,1,1 and F2,2,1 are classified.