Ren Guo
In this paper we characterize hyperbolic geometry among Hilbert geometry by the property that three medians of any hyperbolic triangle all pass through one point.
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The Hilbert metric on convex subsets of $\mathbb R^n$ has proven a rich notion and has been extensively studied. We propose here a generalization of this metric to subset of complex projective spaces and give examples of applications to diverse fields. Basic examples include the classical Hilbert metric which coincides with the hyperbolic metric on real hyperbolic spaces as well as the complex hyperbolic metric on complex hyperbolic spaces.
We prove, in the context of Hilbert geometry, the equivalence between the existence of an upper bound on the area of ideal triangles and the Gromov-hyperbolicity.
We prove that the Hilbert geometry of a convex domain in the plane is Gromov hyperbolic, if, and only if, the bottom of its spectrum is not zero
Tetrahedra in complex hyperbolic space and Hilbert spaces with Pick kernels
本文研究复双曲空间几何与完全Pick核Hilbert空间的关系，提出顶点角余弦的复模拟，用于描述四面体三角面及四维Pick核空间的三维子空间。
We give a trigonometric proof of the Steiner-Lehmus Theorem in hyperbolic geometry. Precisely we show that if two internal bisectors of a triangle on the hyperbolic plane are equal, then the triangle is isosceles.
Using the method of C. Vörös, we establish results in hyperbolic plane geometry, related to triangles and circles. We present a model independent construction for Malfatti's problem and several trigonometric formulas for triangles.
The goal of this paper is to study two basic problems of hyperbolic geometry. The first problem is to compare the hyperbolic and Euclidean distances. The second problem is to find hyperbolic counterparts of some basic geometric constructions such as the construction of the middle point of a hyperbolic geodesic segment. Apollonian circles have a key role in this study.
Consider the Poincare disc model for hyperbolic geometry. In this paper, a convenient computational formula is developed along with an aesthetic geometric interpretation. Two proofs, one geometric and one analytical, of each result are given.
We begin a coarse geometric study of Hilbert geometry. Actually we give a necessary and sufficient condition for the natural boundary of a Hilbert geometry to be a corona, which is a nice boundary in coarse geometry. In addition, we show that any Hilbert geometry is uniformly contractible and with coarse bounded geometry. As a consequence of these we see that the coarse Novikov conjecture holds for a Hilbert geometry with a mild condition. Also we show that the asymptotic dimension of any two-dimensional Hilbert geometry is just two. This implies that the coarse Baum-Connes conjecture holds fo…
In classical Euclidean geometry, there are several equivalent definitions of conic sections. We show that in the hyperbolic plane, the analogues of these same definitions still make sense, but are no longer equivalent, and we discuss the relationships among them.
Given a pair of points in the hyperbolic half plane or the unit disk, we provide a simple construction of the midpoint of the hyperbolic geodesic segment joining the points.
We consider some constructions in hyperbolic geometry that are analogous to classical constructions in Euclidean geometry. We show that both Monge's theorem and the theorem on the concurrence of the common chords of three circles also hold in absolute geometry. We proffer analogues of the Euler line and the Euler nine-point circle and also extend Feuerbach's famous theorem.
We prove in this paper that the Hilbert geometry associated with an open convex polygonal set is Lipschitz equivalent to Euclidean plane.
Assume that M is a closed hyperbolic 3-manifold fibering over the circle with fiber a closed orientable surface of genus g. We show that if M has large diameter and its injectivity radius is bounded below, then the rank of the fundamental group of M is 2g+1.
If there exists a cyclic quadrilateral whose sides go through the given four collinear points, then there are infinitely many such quadrilaterals inscribed in the same circle. We give two proofs of this porism; one based on cross-ratios, the other on compositions of hyperbolic isometries.
The aim of this paper is to provide two examples in Hilbert geometry which show that volume growth entropy is not always a limit on the one hand, and that it may vanish for a non-polygonal domain in the plane on the other hand.
A Poincaré ball model for taxicab hyperbolic geometry
本文将采用替代距离概念的出租车空间与庞加莱球模型合并，创建出租车版庞加莱球，确定其等距群并证明其在格罗莫夫意义下为双曲空间。
We define and study an extended hyperbolic space which contains the hyperbolic space and de Sitter space as subspaces and which is obtained as an analytic continuation of the hyperbolic space. The construction of the extended space gives rise to a complex valued geometry consistent with both the hyperbolic and de Sitter space. Such a construction shed a light and inspires a new insight for the study of the hyperbolic geometry and Lorentzian geometry. We discuss the advantages of this new geometric model as well as some of its applications.
In Euclidean geometry the circle of Apollonious is the locus of points in the plane from which two collinear adjacent segments are perceived as having the same length. In Hyperbolic geometry, the analog of this locus is an algebraic curve of degree four which can be bounded or "unbounded". We study this locus and give a simple description of this curve using the half-plane model. In the end, we give the motivation of our investigation and calculate the probability that three collinear adjacent segments can be seen as of the same positive length under some natural assumptions about the setting …
The horofunction boundary is a means of compactifying metric spaces that was introduced by Gromov in the 1970s. We describe explicitly the horofunction boundary of the Hilbert geometry, and sketch how it may be used to study the isometry group of this space.
We survey the Hilbert geometry of convex polytopes. In particular we present two important characterisations of these geometries, the first one in terms of the volume growth of their metric balls, the second one as a bi-lipschitz class of the simplexe's geometry.
This second part on polygons in the hyperbolic plane is based on the first part which deals with uniqueness and existence of cocyclic polygons with prescribed sidelengths. The topic here is the maximum question for the area of these polygons. It is shown that the maximal copies must be cocyclic and oriented-convex or have a special collinear shape. The maximal copies are unique up to rigid hyperbolic motions. The cocyclicity means that the vertices are situated on a distance circle, a distance line or a horocycle, the three types of cycles in the hyperbolic plane. The phenomenon of different c…
Large-scale sublinearly Lipschitz maps have been introduced by Yves Cornulier in order to precisely state his theorems about asymptotic cones of Lie groups. In particular, Sublinearly biLipschitz Equivalences (SBE) are a weak variant of quasiisometries, with the only requirement of still inducing biLipschitz maps at the level of asymptotic cones. We focus here on hyperbolic metric spaces and study properties of their boundary extensions, reminiscent of quasiM{ö}bius mappings. We give a dimensional invariant on the boundary that allows to distinguish hyperbolic symmetric spaces up to SBE, answe…
The goal of this paper is to introduce and study analogues of the Euclidean Funk and Hilbert metrics on open convex subsets $Ω$ of hyperbolic or spherical spaces. At least at a formal level, there are striking similarities among the three cases: Euclidean, spherical and hyperbolic. We start by defining non-Euclidean analogues of the Euclidean Funk weak metric and we give three distinct representations of it in each of the non-Euclidean cases, which parallel the known situation for the Euclidean case. As a consequence, all of these metrics are shown to be Finslerian, and the associated norms of…
We provide hyperbolic analogues of some classical theorems in spherical geometry due to Menelaus, Euler, Lexell, Ceva and Lambert. Some of the spherical results are also made more precise.
We provide a new formulation and proof of the triangle altitudes theorem in hyperbolic plane geometry, together with an easily computed discriminant to distinguish between different basic configurations of the altitudes of such a triangle.
We describe a hyperbolic version of the Ambartzumian-Pleijel identity. We use this identity to prove the hyperbolic Crofton formula and the hyperbolic isoperimetric inequality. This identity also provides a way to compute the chord length distribution for an ideal polygon in the hyperbolic plane. The analogous results for a maximally symmetric, simply connected, $2$-dimensional Riemannian manifold with constant sectional curvature are given at the end.
This article provides a simple pictorial introduction to universal hyperbolic geometry. We explain how to understand the subject using only elementary projective geometry, augmented by a distinguished circle. This provides a completely algebraic framework for hyperbolic geometry, valid over the rational numbers (and indeed any field not of characteristic two), and gives us many new and beautiful theorems. These results are accurately illustrated with colour diagrams, and the reader is invited to check them with ruler constructions and measurements.
Certain triangle inequalities involving the circumradius, inradius, and side lengths of a triangle are generalized to spherical and hyperbolic geometry. Examples include strengthenings of Euler's inequality, $R\geq2r$. An extension of Euler's inequality to a simplex in $n$-dimensional space is also generalized to spherical geometry.
We investigate several topics of triangle geometry in the elliptic and in the extended hyperbolic plane, such as: centers based on orthogonality, centers related to circumcircles and incircles, radical centers and centers of similitude, orthology, Kiepert perspectors and related objects, Tucker circles, isoptics, substitutes for the Euler line. For both, the elliptic and the extended hyperbolic plane, a uniform metric is used.
A soft presentation of hyperbolic spaces, free of differential apparatus, is offered. Fifth Euclid's postulate in such spaces is overthrown and, among other things, it is proved that spheres (equipped with great-circle distances) and hyperbolic and Euclidean spaces are the only locally compact geodesic (i.e., convex) metric spaces that are three-point homogeneous.
We prove that the Hilbert geometry of a product of convex sets is bi-lipschitz equivalent the direct product of their respective Hilbert geometries. We also prove that the volume entropy is additive with respect to product and that amenability of a product is equivalent to the amenability of each terms.
Hyperbolic geometry is developed in a purely algebraic fashion from first principles, without a prior development of differential geometry. The natural connection with the geometry of Lorentz, Einstein and Minkowski comes from a projective point of view, with trigonometric laws that extend to `points at infinity', here called `null points', and beyond to `ideal points' associated to a hyperboloid of one sheet. The theory works over a general field not of characteristic two, and the main laws can be viewed as deformations of those from planar rational trigonometry. There are many new features.
This is survey about action of group on Hilbert geometry. It will be a chapter of the "Handbook of Hilbert geometry" edited by G. Besson, M. Troyanov and A. Papadopoulos.
It is shown that the Hilbert metric on the interior of a convex polytope is bilipschitz to a normed vector space of the same dimension.
Certain topics on polygons are extended from Euclidean to hyperbolic geometry. This first part deals with uniqueness and existence of cocyclic polygons with prescribed sidelengths. The non-Euclidean versions are more difficult due to the existence of three different types of circles in the hyperbolic plane. The second part will be concerned with the problem of maximizing the area of polygons with fixed sidelengths. By the metric character of these problems it is necessary to dispense to a certain extent with classical hyperbolic trigonometry because angles must be replaced by length quantities…
Motivated by problems of hyperbolic stochastic geometry we introduce and study the class of beta-star polytopes. A beta-star polytope is defined as the convex hull of an inhomogeneous Poisson processes on the complement of the unit ball in $\mathbb{R}^d$ with density proportional to $(||x||^2-1)^β$, where $||x|| > 1$ and $β>d/2$. Explicit formulas for various geometric and combinatorial functionals associated with beta-star polytopes are provided, including the expected number of $k$-dimensional faces, the expected external angle sums and the expected intrinsic volumes. Beta-star polytopes are…
Consider a stationary Poisson process in a $d$-dimensional hyperbolic space. For $R>0$ define the point process $ξ_R^{(k)}$ of exceedance heights over a suitable threshold of the hyperbolic volumes of $k$th nearest neighbour balls centred around the points of the Poisson process within a hyperbolic ball of radius $R$ centred at a fixed point. The point process $ξ_R^{(k)}$ is compared to an inhomogeneous Poisson process on the real line with intensity function $e^{-u}$ and point process convergence in the Kantorovich-Rubinstein distance is shown. From this, a quantitative limit theorem for the …
We show that the spheres in Hilbert geometry have the same volume growth entropy as those in the Lobachevsky space. We give the asymptotic estimates for the ratio of the volume of metric ball to the area of the metric sphere in Hilbert geometry. Derived estimates agree with the well-known fact in the Lobachevsky space
In this note, we will explain the connection between the Seven Circles Theorem and hyperbolic geometry, then prove a stronger result about hyperbolic geometry hexagons which implies the Seven Circles Theorem as a special case.
We establish some geometric constraints on compact Coxeter polytopes in hyperbolic spaces and show that these constraints can be a very useful tool for the classification problem of reflective anisotropic Lorentzian lattices and cocompact arithmetic hyperbolic reflection groups.
In this article, we study the geometry of an infinite dimensional Hyperbolic space. We will consider the group of isometries of the Hilbert ball equipped with the Carath$\acute{e}$odory metric and learn about some special subclasses of this group. We will also find some unitary equivalence condition and compute some cardinalities.
The memoir Theorie der Parallellinien (1766) by Johann Heinrich Lambert is one of the founding texts of hyperbolic geometry, even though its author's aim was, like many of his pre-decessors', to prove that such a geometry does not exist. In fact, Lambert developed his theory with the hope of finding a contradiction in a geometry where all the Euclidean axioms are kept except the parallel axiom and that the latter is replaced by its negation. In doing so, he obtained several fundamental results of hyperbolic geometry. This was sixty years before the first writings of Lobachevsky and Bolyai appe…
H. L. Skala (1992) gave the first elegant first-order axiom system for hyperbolic geometry by replacing Menger's axiom involving projectivities with the theorems of Pappus and Desargues for the hyperbolic plane. In so doing, Skala showed that hyperbolic geometry is incidence geometry. We improve upon Skala's formulation by doing away with Pappus and Desargues altogether, by substituting for them two simpler axioms.
Let $τ$ be an $n$-simplex and let $g$ be a metric on $τ$ with constant curvature $κ$. The lengths that $g$ assigns to the edges of $τ$, along with the value of $κ$, uniquely determine all of the geometry of $(τ, g)$. In this paper we focus on hyperbolic simplices ($κ= -1$) and develop geometric formulas which rely only on the edge lengths of $τ$. Our main results are distance and projection formulas in hyperbolic simplices, as well as a projection formula in Euclidean simplices. We also provide analogous formulas in simplices with arbitrary constant curvature $κ$.
The aim of this article is to understand the geometry of limit sets in pseudo-Riemannian hyperbolic geometry. We focus on a class of subgroups of $\mathrm{PO}(p,q+1)$ introduced by Danciger, Guéritaud and Kassel, called $\mathbb{H}^{p,q}$-convex cocompact. We define a pseudo-Riemannian analogue of critical exponent and Hausdorff dimension of the limit set. We show that they are equal and bounded from above by the usual Hausdorff dimension of the limit set. We also prove a rigidity result in $\mathbb{H}^{2,1}=\mathrm{ADS}^3$ which can be understood as a Lorentzian version of a famous Theorem of…
The main result is that every complete finite area hyperbolic metric on a sphere with punctures can be uniquely realized as the induced metric on the surface of a convex ideal polyhedron in hyperbolic 3-space. A number of other observations are included.
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