Study angle structures on pseudo 3-manifolds, proving existence for some cases.
problem Determining if hyperbolic 3-manifolds can have angle structures.
method Examined triangulated pseudo 3-manifolds with area-curvature angle structures, establishing sufficient and necessary conditions.
result Compact hyperbolic 3-manifolds with totally geodesic boundary can have angle structures.
The paper examines rigidity in geometric actions of Coxeter groups on Croke-Kleiner spaces.
problem The rigidity of geometric actions of Coxeter groups compared to their quasi-isometric counterparts.
method Study of right-angled Coxeter groups acting geometrically on Croke-Kleiner spaces.
result Right-angled Coxeter groups have more rigid geometric actions than their quasi-isometric counterparts.
The paper solves a geometric problem involving points in a triangle's plane.
problem Determine points in a triangle's plane corresponding to given cosines of angles.
method Analyzes the geometric constraints and uses trigonometric properties.
result The number of points D satisfying the given conditions is determined. Smooth approximations bound dihedral angles of convex polytopes.
problem Bounding dihedral angles of convex polytopes.
method Approximating polytopes with smooth hypersurfaces and using geometric relations.
result Established lower bounds on dihedral angles.
Study rigidity of real moment-angle manifolds using cubical geometry.
problem Topological rigidity of real moment-angle manifolds.
method Cubical geometry and surgery theory.
result Real moment-angle manifolds of dimension at least five satisfy the Borel Conjecture.
Fixed angles of convex polygons lead to combinatorially rich polytopes.
problem Understanding the structure of convex polygons with fixed vertex angles.
method Combining combinatorial and geometric approaches, including dual polytopes and Schwarz-Christoffel maps.
result Fixed-angles polytopes are dual to cyclic polytopes under certain conditions.
We describe some properties of noncompact Euclidean cone manifolds with cone angles less than c less than 2pi and singular locus a submanifold. More precisely, we describe its structure outside a compact set. As a corollary we classify those with cone angles less than 3pi/2 and those with all cone angles equal to 3pi/2…
Investigates polar tangential angles of curves and their monotonicity.
problem Behavior of polar tangential angles of plane curves.
method Proof of monotonicity for certain curves of monotone curvature.
result Nonexistence results for an obstacle problem involving free elasticae.
Moment-angle manifolds provide a wide class of examples of non-Kaehler compact complex manifolds. A complex moment-angle manifold Z is constructed via certain combinatorial data, called a complete simplicial fan. In the case of rational fans, the manifold Z is the total space of a holomorphic bundle over a toric variet…
The paper studies the face angles of tetrahedra with a fixed base.
problem Determine the closure and boundary of the set of face angles of tetrahedra with a given base.
method Analyzes the set of tetrahedra with a given base and calculates the cosine of the angles between the faces.
result The closure and boundary of the set of face angles are determined.
We prove the strong Atiyah conjecture for right-angled Artin groups and right-angled Coxeter groups. More generally, we prove it for groups which are certain finite extensions or elementary amenable extensions of such groups.
New theorem disproves Angle Defect for super triangles.
problem Angle Defect Theorem for N=1 super hyperbolic geometry.
method Action of OSp(1|2) on real super Minkowski space and brute-force computation.
result Disproves Angle Defect Theorem and provides novel additive function.
We investigate the planarity of the boundaries of right-angled Coxeter groups. We show that non-planarity of the defining graph does not necessarily imply non-planarity of every boundary of the associated right-angled Coxeter group, although it does in many cases. Our techniques yield a characterization of the triangle…
New rigidity results for complex and quaternionic moment-angle manifolds.
problem Equivariant topological rigidity of complex and quaternionic moment-angle manifolds.
method Reduction to equivariant rigidity of quasitoric (or quoric) quotients and principal bundles.
result Full equivariant rigidity for manifolds with four-dimensional quoric quotients and primary rigidity for higher dimensions.
Defines Kahler angle for a broader context.
problem Generalizing results about Kahler angle.
method Provides a general definition of Kahler angle.
result Generalized results about Kahler angle.
We first define a complex angle between two oriented spacelike planes in 4-dimensional Minkowski space, and then study the constant angle surfaces in that space, i.e. the oriented spacelike surfaces whose tangent planes form a constant complex angle with respect to a fixed spacelike plane. This notion is the natural Lo…
We discuss the existence of the angle between two curves in Teichmüller spaces and show that, in any infinite dimensional Teichmüller space, there exist infinitely many geodesic triangles each of which has the same three vertices and satisfies the property that its three sides have the same and arbitrarily given length…
The study examines subgroups of RACGs and RAAGs, focusing on their RAAG properties.
problem Characterizing subgroups of right-angled Coxeter and Artin groups that are themselves RAAGs.
method Analyzes specific classes of subgroups and uses quasi-isometry and commensurability properties.
result Characterizes finite-index visual RAAG subgroups of 2-dimensional RACGs and provides new examples of RACGs commensurable to RAAGs.
Researchers solved a geometry paradox for creased tubes.
problem Resolving the paradox of Gaussian curvature in creased tubes.
method Calculated Gaussian curvature in terms of rate of change of solid angle, dependent on fold angle and curvature.
result Gaussian curvature is zero overall despite the surface being doubly-curved.
Introduces a new geometry based on difference angles, showing unique properties.
problem Defining angles independently of circles or rotations.
method Axiomatic system for difference angles, defining new geometric constructs.
result Explicit confirmation of the concurrency of the parabolic Miquel configuration.
Stiefel-Whitney classes of moment-angle manifolds are trivial.
problem Analyzing the topological properties of moment-angle manifolds.
method Proving triviality of Stiefel-Whitney classes for moment-angle manifolds, including partial quotients.
result Stiefel-Whitney classes of moment-angle manifolds are trivial.
This article defines a pair of combinatorial operations on the combinatorial structure of compact right-angled hyperbolic polyhedra in dimension three called decomposition and edge surgery. It is shown that these operations simplify the combinatorics of such a polyhedron, while keeping it within the class of right-angl…
In this paper, we discuss the Lagrangian angle and the Kähler angle of immersed surfaces in C2. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in C2 than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surface…
A novel method optimizes variable-stiffness structures for better strength and weight.
problem Optimizing variable-stiffness structures for higher strength and lighter weight.
method A novel multi-stage concurrent topology optimization scheme combining DMO, S-BPTO, and CFAO.
result The method ensures better fibre angle convergence and stable optimization.
Classifies divergence and thickness in right-angled Coxeter groups.
problem Characterizing the divergence and thickness of right-angled Coxeter groups.
method Completely classifies divergence functions and proves conditions for thickness using the hypergraph index.
result Exact divergence functions of RACGs can be computed from their defining graphs.
Optical interpretation of Euler's angle problem for caustics of light rays.
problem Euler's angle problem for caustics of light rays.
method Optical interpretation of intrinsic equations and study of caustics.
result Characterization of curves under specific conditions and novel pantograph equations.
As was pointed out by Nikulin [8] and Vinberg [10], a right-angled polyhedron of finite volume in hyperbolic n-space Hn has at least one cusp for n≥5. We obtain non-trivial lower bounds on the number of cusps of such polyhedra. For example, right-angled polyhedra of finite volume must have at least th…
The paper explores connections between perimeter, area, and visual angle of convex sets.
problem Understanding geometric properties of convex sets through visual angle and related measurements.
method Establishing universal formulas and characterizing convex sets of constant width.
result Crofton's formula is the unique universal formula relating visual angle, length, and area.
Uniqueness of quasi-roots explored in right-angled Artin groups.
problem Uniqueness of quasi-roots in right-angled Artin groups.
method Introducing quasi-roots and studying their uniqueness.
result Uniqueness of quasi-roots established in right-angled Artin groups.
We determine the factorial growth rate of the number of finite index subgroups of right-angled Artin groups as a function of the index. This turns out to depend solely on the independence number of the defining graph. We also make a conjecture for right-angled Coxeter groups and prove that it holds in a limited setting…
We give a necessary and sufficient condition for a graph to have a right-angled Artin group as its braid group for braid index ≥5. In order to have the necessity part, graphs are organized into small classes so that one of homological or cohomological characteristics of right-angled Artin groups can be applied. Fi…
We develop an analogy between right-angled Artin groups and mapping class groups through the geometry of their actions on the extension graph and the curve graph respectively. The central result in this paper is the fact that each right-angled Artin group acts acylindrically on its extension graph. From this result we …
Study on Laplacian determinant in isosceles triangles, finding equilateral triangle minimizes determinant.
problem Finding the minimum of the spectral determinant on isosceles triangles.
method Analyzing the determinant of the Laplacian on Euclidean isosceles triangle envelopes of fixed area.
result Equilateral triangle envelope minimizes the determinant of the Laplacian.
Study of curves and surfaces in Riemannian spaces making a constant angle with a parallel transported direction.
problem Understanding geometric properties of curves and surfaces in Riemannian spaces.
method Developing a theoretical framework to study curves and surfaces by their angle with a parallel transported vector field.
result Surfaces making a constant angle with a parallel transported direction are extrinsically flat ruled surfaces.
This note generalizes the visual angle to convex sets in 3D space.
problem Analyzing geometric properties of convex sets in 3D space.
method Generalizing the visual angle to convex sets in Euclidean space and expressing geometric quantities in terms of integrals of functions related to the solid angle.
result Invariant quantities of the original convex set can be expressed by integrals of functions related to the solid angle.
Study proves existence of weak mean curvature flow with contact angle.
problem Existence of weak mean curvature flow with prescribed contact angle.
method Compactness theorem for varifolds and Ilmanen's regularization extended to capillarity.
result Existence of weak mean curvature flow with contact angle for general θ. Improved volume estimates for right-angled polyhedra in hyperbolic space.
problem Estimating volumes of right-angled polyhedra in hyperbolic space.
method Combining Andreev theorem and Atkinson's results, improved upper volume estimates.
result Upper volume estimates for both compact and ideal right-angled polyhedra improved.
This paper explores historical and philosophical aspects of angles and solid angles, inspired by Euler's work.
problem Understanding the historical context and philosophical implications of angles and solid angles.
method Historical review and analysis of mathematical and philosophical works.
result Questions raised by Euler about angles and solid angles are timeless and relevant to modern mathematics.
Real moment-angle manifolds of combinatorially equivalent simple polytopes are equivariantly diffeomorphic.
problem Uniqueness of smooth structures on real moment-angle manifolds.
method Arguments from calculus applied to results from complex moment-angle manifolds.
result Real moment-angle manifolds of combinatorially equivalent simple polytopes are equivariantly diffeomorphic.
Improved bounds on acylindricity for right-angled Artin groups.
problem Bounding the acylindrical action of right-angled Artin groups on their extension graphs.
method Exploring lattice properties, studying prefixes of powers, and extending quasi-root uniqueness.
result Cardinality of r-quasi-stabilizer is bounded by a linear function of r. New invariant from links to polyhedra volumes.
problem Computing hyperbolic volumes of link complements.
method Geometric, topological, and combinatorial methods to decompose link complements into ideal polyhedra.
result A new geometric link invariant, the right-angled volume, is a lower bound for hyperbolic volume.
We show that every graph product of finitely generated abelian groups acts properly and cocompactly on a CAT(0) cubical complex. The complex generalizes (up to subdivision) the Salvetti complex of a right-angled Artin group and the Coxeter complex of a right-angled Coxeter group. In the right-angled Artin group case it…
Agol recently introduced the notion of a veering triangulation, and showed that such triangulations naturally arise as layered triangulations of fibered hyperbolic 3-manifolds. We prove, by a constructive argument, that every veering triangulation admits positive angle structures, recovering a result of Hodgson, Rubins…
We provide a congruence theorem for minimal surfaces in S5 with constant contact angle using Gauss-Codazzi-Ricci equations. More precisely, we prove that Gauss-Codazzi-Ricci equations for minimal surfaces in S5 with constant contact angle satisfy an equation for the Laplacian of the holomorphic angle. Also, we wi…
New right-angled Artin subgroups found in Artin groups.
problem Finding large right-angled Artin subgroups in Artin groups.
method Examining centers of irreducible spherical special subgroups and their powers.
result Conjecture verified for certain classes of Artin groups, leading to hyperbolic surface subgroup conclusions.
Surveying connections between graph combinatorics and algebraic right-angled Artin groups.
problem Understanding the relationship between graph structures and algebraic properties of right-angled Artin groups.
method Analyzing the defining and extension graphs of right-angled Artin groups.
result Discovers connections to geometric group theory and complexity theory.
We give a new notion of angle in general metric spaces; more precisely, given a triple a points p,x,q in a metric space (X,d), we introduce the notion of angle cone ∠pxq as being an interval ∠pxq:=[∠pxq−,∠pxq+], where the quantities ∠pxq± are defined in terms o…
The paper studies prescribed angle surfaces in Riemannian manifolds with torse-forming vector fields.
problem Characterizing surfaces with prescribed angles in Riemannian geometry.
method Introducing and analyzing prescribed angle hypersurfaces associated with torse-forming vector fields.
result Classification of prescribed angle surfaces in 3D Riemannian manifolds.