arXiv research

Paper extends circle pattern theory to obtuse angles.

Paper generalizes Andreev's theorem with obtuse angles.

Thurston's Circle Pattern Theorem studies existence and rigidity of circle patterns of a given combinatorial type and the given non-obtuse exterior intersection angles. Using topological degree theory, variational principle, Teichmuller theory, and Sard's Theorem, this paper generalizes Circle Pattern Theorem to the ca…

The study finds all possible 3D polytopes in Riemannian 3-manifolds with positive scalar curvature.

Given a combinatorial description $C$ of a polyhedron having $E$ edges, the space of dihedral angles of all compact hyperbolic polyhedra that realize $C$ is generally not a convex subset of $\mathbb{R}^E$ \cite{DIAZ}. If $C$ has five or more faces, Andreev's Theorem states that the corresponding space of dihedral angle…

We give a method for computing upper and lower bounds for the volume of a non-obtuse hyperbolic polyhedron in terms of the combinatorics of the 1-skeleton. We introduce an algorithm that detects the geometric decomposition of good 3-orbifolds with planar singular locus and underlying manifold the 3-sphere. The volume b…

We demonstrate how to construct three-dimensional compact hyperbolic polyhedra using Newton's Method. Under the restriction that the dihedral angles are non-obtuse, Andreev's Theorem provides as necessary and sufficient conditions five classes of linear inequalities for the dihedral angles of a compact hyperbolic polyh…

The paper extends the Discrete Schwarz-Pick Lemma to circle packings with obtuse intersections and disjoint packings.

This paper investigates circle patterns with obtuse exterior intersection angles on surfaces of finite topological type. We characterise the images of the curvature maps and establish several equivalent conditions regarding long time behaviors of Chow-Luo's combinatorial Ricci flows for these patterns. As consequences,…

We introduce a new geometric invariant called the obtuse constant of spaces with curvature bounded below. We first find relations between this invariant and the normalized volume. We also discuss the case of maximal obtuse constant equal to $π/2$, where we prove some rigidity for spaces. Although we consider Alexandrov…

In 1970, E. M. Andreev published a classification of all three-dimensional compact hyperbolic polyhedra having non-obtuse dihedral angles. Given a combinatorial description of a polyhedron, $C$, Andreev's Theorem provides five classes of linear inequalities, depending on $C$, for the dihedral angles, which are necessar…

The paper studies how spaces collapse to Alexandrov spaces with mild singularities.

Study on folded ribbon knots and their minimum length.

Trapezoids uniquely identified by their Dirichlet Laplace spectrum.

We prove that a 3--dimensional hyperbolic cusp with convex polyhedral boundary is uniquely determined by its Gauss image. Furthermore, any spherical metric on the torus with cone singularities of negative curvature and all closed contractible geodesics of length greater than $2π$ is the metric of the Gauss image of som…

Introduces Coxeter polyhedra in various geometries.

We study the problem of determining the least symmetric triangle, which arises both from pure geometry and from the study of molecular chirality in chemistry. Using the correspondence between planar $n$-gons and points in the Grassmannian of 2-planes in real $n$-space introduced by Hausmann and Knutson, this correspond…

We consider the problem of finding the probability that a random triangle is obtuse, which was first raised by Lewis Caroll. Our investigation leads us to a natural correspondence between plane polygons and the Grassmann manifold of 2-planes in real $n$-space proposed by Allen Knutson and Jean-Claude Hausmann. This cor…

From a simple observation about a construction of Thurston, we derive several interesting facts about subgroups of the mapping class group generated by two positive multi-twists. In particular, we identify all configurations of curves for which the corresponding groups fail to be free, and show that a subset of these d…

Defines Kahler angle for a broader context.

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…

Study angle structures on pseudo 3-manifolds, proving existence for some cases.

In this paper we study the right-angled Coxeter groups that acts geometrically on the Salvetti complex of a certain right-angled Artin group, which we refer to as Croke-Kleiner spaces. We prove that any right-angled Coxeter group that acts geometrically on the Croke-Kleiner spaces acts with $π/2$ angles between reflect…

Introduces a new geometry based on difference angles, showing unique properties.

Stiefel-Whitney classes of moment-angle manifolds are trivial.

In this paper, we discuss the Lagrangian angle and the Kähler angle of immersed surfaces in $\mathbb C^2$. Firstly, we provide an extension of Lagrangian angle, Maslov form and Maslov class to more general surfaces in $\mathbb C^2$ than Lagrangian surfaces, and then naturally extend a theorem by J.-M. Morvan to surface…

Uniqueness of quasi-roots explored in right-angled Artin groups.

This note generalizes the visual angle to convex sets in 3D space.

Study proves existence of weak mean curvature flow with contact angle.

Improved volume estimates for right-angled polyhedra in hyperbolic space.

This paper explores historical and philosophical aspects of angles and solid angles, inspired by Euler's work.

Real moment-angle manifolds of combinatorially equivalent simple polytopes are equivariantly diffeomorphic.

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 $S^5$ with constant contact angle using Gauss-Codazzi-Ricci equations. More precisely, we prove that Gauss-Codazzi-Ricci equations for minimal surfaces in $S^5$ with constant contact angle satisfy an equation for the Laplacian of the holomorphic angle. Also, we wi…

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…

Surveying connections between graph combinatorics and algebraic right-angled Artin groups.

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 ${\angle_{pxq}}$ as being an interval ${\angle_{pxq}}:=[\angle^-_{pxq},\angle^+_{pxq}]$, where the quantities $\angle^\pm_{pxq}$ are defined in terms o…

The paper studies prescribed angle surfaces in Riemannian manifolds with torse-forming vector fields.

In this paper we classify certain special ruled surfaces in $\R^3$ under the general theorem of characterization of constant angle surfaces. We study the tangent developable and conical surfaces from the point of view the constant angle property. Moreover, the natural extension to normal and binormal constant angle sur…

We survey the role of right-angled Artin groups in the theory of diffeomorphism groups of low dimensional manifolds. We first describe some of the subgroup structure of right-angled Artin groups. We then discuss the interplay between algebraic structure, compactness, and regularity for group actions on one--dimensional…

Study on null hypersurfaces with constant angle in Lorentzian manifolds.

Establishes a boundary maximum principle for varifolds with fixed contact angle.

This notes explores angle structures on ideally triangulated compact $3$-manifolds with high genus boundary. We show that the existence of angle structures implies the existence of a hyperbolic metric with totally geodesic boundary, and conversely each hyperbolic $3$-manifold with totally geodesic boundary has an ideal…

The paper solves a geometric problem involving points in a triangle's plane.

The paper studies the face angles of tetrahedra with a fixed base.

We consider the question of determining whether a given group (especially one generated by involutions) is a right-angled Coxeter group. We describe a group invariant, the involution graph, and we characterize the involution graphs of right-angled Coxeter groups. We use this characterization to describe a process for c…

In Lorentzian geometry, limited definition of angles restricts the use of angle bisectors in study of triangles. This paper redefines angle bisectors so that they can be used to study attributes of triangles. Using the new definition, this paper investigates the existence of the incenter and the isogonal conjugate of a…

Hyperbolic links in thickened torus decompose into angled tetrahedra.