arXiv research

Study calculates eigenvalues and eigenfunctions for spherical triangles and finds fundamental gap behavior.

Describes the space of spherical triangles on a smooth 3-manifold.

Napoleonic triangles don't exist in hyperbolic geometry.

We prove that the cosine law for spherical triangles and spherical tetrahedra defines integrable systems, both in the sense of multidimensional consistency and in the sense of dynamical systems.

3-manifolds are CR uniformized on spheres, proving a conjecture.

A spherical polyhedron surface is a triangulated surface obtained by isometric gluing of spherical triangles. For instance, the boundary of a generic convex polytope in the 3-sphere is a spherical polyhedron surface. This paper investigates these surfaces from the point of view of inner angles. A rigidity result is obt…

We establish the equivalence of the Tuynman midpoint area formula for a spherical triangle to the classical area formulas of Euler and of Cagnoli. The derivation also yields a variant of the Cagnoli formula in terms of the medial triangle. We introduce the three barycentric coordinates of a point within the spherical t…

Complex hyperbolic triangle groups yield specific 3-manifolds at infinity.

In this paper we describe trigonometry on the de Sitter surface. For that a characterization of geodesics is given, leading to various types of triangles. We define lengths and angles of these. Then, transferring the concept of polar triangles from spherical geometry into the Minkowski space, we relate hyperbolic with …

We describe the set of possible vector valued side lengths of n-gons in thick Euclidean buildings of rank 2. This set is determined by a finite set of homogeneous linear inequalities, which we call the generalized triangle inequalities. These inequalities are given in terms of the combinatorics of the spherical Coxeter…

The abstract proves spherical surface decompositions with conical singularities.

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…

New CR representations are found and shown to be redundant.

The Apollonius theorem is generalized for m-simplices, with applications in geometry and optimization.

The paper explores geometric properties of interception curves on planes and spheres.

Let $M$ be either the 2-sphere $\SS^2 \subset\RR^3$ or the hyperbolic plane $\HH^2 \subset \RR^3$. If $Δ(abc)$ is a geodesic triangle on $M$ with corners at $a,b,c\in M$, we denote by $α, β, γ\in M$ the midpoints of their sides. If $Ω$ denotes the oriented area of this triangle on $M$, it satisfies the relations: $$ \s…

Uniformizes CR structure on a specific 3-manifold.

We develop a transitional geometry, that is, a family of geometries of constant curvatures which makes a continuous connec-tion between the hyperbolic, Euclidean and spherical geometries. In this transitional setting, several geometric entities like points, lines, dis-tances, triangles, angles, area, curvature, etc. as…

Euler explored spherical geometry using trigonometric formulae and solid geometry methods.

We are interested in the 3-Calabi-Yau categories $\mathcal{D}$ arising from quivers with potential associated to a triangulated marked surface $\mathbf{S}$ (without punctures). We prove that the spherical twist group ST of $\mathcal{D}$ is isomorphic to a subgroup (generated by braid twists) of the mapping class group …

Study on spherical bodies of constant width on the unit sphere, proving bounds on their relative effective radius.

The tilings of the 2-dimensional sphere by congruent triangles have been extensively studied, and the edge-to-edge tilings have been completely classified. However, not much is known about the tilings by other congruent polygons. In this paper, we classify the simplest case, which is the edge-to-edge tilings of the 2-d…

Characterizes Lamé equations with finite monodromy on flat tori.

New model for Calabi-Yau-$\mathbb{X}$ categories using decorated marked surfaces.

New bounds on inscribed triangles in arbitrary planar domains.

Paper calculates eigenvalues of a specific triangle on a sphere.

Study on Laplacian determinant in isosceles triangles, finding equilateral triangle minimizes determinant.

New method shows any triangle group generating pair is related to special coverings.

Complex hyperbolic triangle groups were first considered by Mostow in building the first nonarithmetic lattices in PU(2, 1). They are a natural generalization of the classical triangle groups acting on the hyperbolic plane. A well-known theorem of Takeuchi is that there are only finitely many Fuchsian triangle groups t…

Study reveals hidden accidental parabolics in complex hyperbolic geometry.

Defines band maps in unoriented link Floer homology forming a skein exact triangle.

Proves a new skein exact triangle for real monopole Floer homology.

New surgery exact triangles in Heegaard Floer homology for rational slopes.

We use geometric techniques to explicitly find the topological structure of the space of SO(3)-representations of the fundamental group of a closed surface of genus 2 quotient by the conjugation action by SO(3). There are two components of the space. We will describe the topology of both components and describe the cor…

We answer the question "Does the Y-triangle move preserve intrinsic knottedness?" in the negative by giving an example of a graph that is obtained from the intrinsically knotted graph K_7 by triangle-Y and Y-triangle moves but is not intrinsically knotted.

We show that the triangle with angles Pi/12, Pi/3 and 7*Pi/12 has the lattice property and compute this triangle's Veech group.

We study side-lengths of triangles in path metric spaces. We prove that unless such a space X is bounded, or quasi-isometric to line or half-line, every triple of real numbers satisfying the strict triangle inequalities, is realized by the side-lengths of a triangle in X. We construct an example of a complete path metr…

Two triples of triangles having pairwise disjoint outlines in 3-space are called combinatorially isotopic if one triple can be obtained from the other by a continuous motion during which the outlines of the triangles remain pairwise disjoint. We conjecture that it can be algorithmically checked if an (ordered or unorde…

New method proves mateability of triangle groups with Blaschke products.

In this paper we study the area of ideals triangles in a convex domain with its Hilbert geometry. We obtain a characterization of the hyperbolic geometry among all the Hilbert geometry in terms of area of ideals triangles. We also obtain a sharp lower bound on the hilbert area of ideal triangles, independant of the con…

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…

A formula for Rademacher symbols in triangle groups is provided.

Study stabilizers of complex hyperbolic triangle groups, finding generators and signatures.

Criterion for stopping conjugacy class enumeration in triangle groups.

New skein exact triangles for link Floer homology.

New theorem disproves Angle Defect for super triangles.

The study examines Lipschitz normally embedded Hölder triangles in 4D space.

Triangle Artin groups split as graphs of free groups under specific conditions.