Jun Murakami, Akira Ushijima
We give a closed formula for volumes of generic hyperbolic tetrahedra in terms of edge lengths. The cue of our formula is by the volume conjecture for the Turaev-Viro invariant of closed 3-manifolds, which is defined from the quantum 6j-symbols. This formula contains the dilogarithm functions, and we specify the adequate branch to get the actual value of the volumes.
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In this paper we present a necessary conditions, that simple close geodesics on regular tetrahedra in the 3-dimensional hyperbolic space must satisfy. Furthermore, we explicitly describe three classes of simple closed geodesics on regular tetrahedra in the hyperbolic 3-space. These are so-called 2-homogeneous, 3-homogeneous and (3,2)-homogeneous geodesics. Up to a rigid motion of a tetrahedron there exists a unique geodesic in each class.
We classify Coxeter decompositions of hyperbolic tetrahedra, i.e. simplices in the hyperbolic space H^3. The paper completes the classification of Coxeter decompositions of hyperbolic simplices.
We give a rigorous geometric proof of the Murakami-Yano formula for the volume of a hyperbolic tetrahedron. In doing so, we are led to consider generalized hyperbolic tetrahedra, which are allowed to be non-convex, and have vertices `beyond infinity'; and we uncover a group, which we call 22.5K, of 23040 scissors-class-preserving symmetries of the space of (suitably decorated) generalized hyperbolic tetrahedra. The group 22.5K contains the Regge symmetries as a subgroup of order 144. From a generic tetrahedron, 22.5K produces 30 distinct generalized tetrahedra in the same scissors class, inclu…
A generalized hyperbolic tetrahedra is a polyhedron (possibly non-compact) with finite volume in hyperbolic space, obtained from a tetrahedron by the polar truncation at the vertices lying outside the space. In this paper it is proved that a volume formula for ordinary hyperbolic tetrahedra devised by J. Murakami and M. Yano can be applied to such ones. There are two key tools for the proof; one is so-called Schlafli's differential formula for hyperbolic polyhedra, and the other is a necessary and sufficient condition for given numbers to be the dihedral angles of a generalized hyperbolic simp…
We define a volume function on configurations of four points in the 3-sphere which is invariant under the action of PU(2,1), the automorphism group of the CR structure defined on the sphere by its embedding in complex 2-space. We show that the volume function, constructed using appropriate combinations of the dilogathm function of Bloch-Wigner, satisfies a five term relation in a more general context which includes at the same time CR and real hyperbolic geometry.
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。
Define a body A to be able to hide behind a body B if the orthogonal projection of B contains a translation of the corresponding orthogonal projection of A in every direction. In two dimensions, it is easy to observe that there exist two objects such that one can hide behind another and have a larger area than the other. It was recently shown that similar examples exist in higher dimensions as well. However, the highest possible volume ratio for such bodies is still undetermined. We investigated two three-dimensional examples, one involving a tetrahedron and a ball and the other involving a te…
We study spherical tetrahedra with rational dihedral angles and rational volumes. Such tetrahedra occur in the Rational Simplex Conjecture by Cheeger and Simons, and we supply vast families, discovered by computational efforts, of positive examples that confirm this conjecture. As a by-product, we also obtain a classification of all spherical Pythagorean triples, previously found by Smith.
Hyperideal tetrahedra are the fundamental building blocks of hyperbolic 3-manifolds with geodesic boundary. The study of their geometric properties (in particular, of their volume) has applications also in other areas of low-dimensional topology, like the computation of quantum invariants of 3-manifolds and the use of variational methods in the study of circle packings on surfaces. The Schläfli formula neatly describes the behaviour of the volume of hyperideal tetrahedra with respect to dihedral angles, while the dependence of volume on edge lengths is worse understood. In this paper we prove …
Trigonometry of partially truncated triangles and tetrahedra
本文利用双曲面模型和洛伦兹几何，建立了具有理想顶点的四边形和五边形的双曲余弦与正弦定理，并证明了部分截断四面体的横向长度仅取决于内部边长集合，且给出了界限。
Given five points in a three-dimensional euclidean space, one can consider five tetrahedra, using those points as vertices. We present a pentagon-like formula containing the product of three volumes of those tetrahedra in its l.h.s. and the product of the two remaining tetrahedron volumes in its r.h.s., as well as the derivative of the "discrete curvature" which arises when we slightly deform our euclidean space.
In this paper we give an affirmative answer to the following question posed by Daryl Cooper: If one lengthens the sides of a tetrahedron by one unit, is the result still a tetrahedron and (if so) does the volume increase? Our proof involves a (presumably) new and sharp inequality involving the Cayley-Menger determinant and one of its directional derivatives. We give a rigorous computer-assisted proof of the inequality. We also sketch an argument which derives the existence portion of the result, in all dimensions, from an old theorem of Von Neumann. Finally, we prove a number of additional res…
We prove elegant trilinear formulas connecting products of volumes of Euclidean tetrahedra with vertices taken from a given set of 6 points. We propose a way for generalizing those formulas.
This paper collects some important formulas on hyperbolic volume. To determine concrete values of volume function is a very hard question requiring the knowledge of various methods. Our goal to give a non-elementary integral on the volume of the orthosceme (obtain it without using the Schläfli differential formula), using edge-lengthes as the only parameters.
We consider the projectivization of Minkowski space with the analytic continuation of the hyperbolic metric and call this an extended hyperbolic space. We can measure the volume of a domain lying across the boundary of the hyperbolic space using an analytic continuation argument. In this paper we show this method can be further generalized to find the volume of a domain with smooth boundary with suitable regularity in dimension 2 and 3. We also discuss that this volume is invariant under the group of hyperbolic isometries and that this regularity condition is sharp.
We show that the 1-cusped quotient of the hyperbolic space $\mathbb{H}^3$ by the tetrahedral Coxeter group $Γ_*=[5,3,6]$ has minimal volume among all non-arithmetic cusped hyperbolic 3-orbifolds, and as such it is uniquely determined. Furthermore, the lattice $Γ_*$ is incommensurable to any Gromov-Piatetski-Shapiro type lattice. Our methods have their origin in the work of C. Adams. We extend considerably this approach via the geometry of the underlying horoball configuration induced by a cusp.
Volumes of Regular Hyperbolic Simplices
本文推导了任意维数双曲空间中正则单形体积的显式公式，为相关几何研究提供了基础工具。
Volume is a natural measure of complexity of a Riemannian manifold. In this survey, we discuss the results and conjectures concerning n-dimensional hyperbolic manifolds and orbifolds of small volume.
We compute the hyperbolic covolume of the automorphism group of each even unimodular Lorentzian lattice. The result is obtained as a consequence of a previous work with Belolipetsky, which uses Prasad's volume to compute the volumes of the smallest hyperbolic arithmetic orbifolds.
We conjecture that for every dimension n not equal 3 there exists a noncompact hyperbolic n-manifold whose volume is smaller than the volume of any compact hyperbolic n-manifold. For dimensions n at most 4 and n=6 this conjecture follows from the known results. In this paper we show that the conjecture is true for arithmetic hyperbolic n-manifolds of dimension n at least 30.
Convex Hull Volumes in Hyperbolic 3-Space
研究双曲三维空间中闭集的几何条件，证明其凸包体积无限，回答Danny Calegari问题，并刻画平面自相似集的几何特性。
We prove that the covolume of any quasi-arithmetic hyperbolic lattice (a notion that generalizes the definition of arithmetic subgroups) is a rational multiple of the covolume of an arithmetic subgroup. As a corollary, we obtain a good description for the shape of the volumes of most of the known hyperbolic n-manifolds with n>3.
In this paper we obtain new upper bounds on volumes of right-angled polyhedra in hyperbolic space $\mathbb{H}^3$ in three different cases: for ideal polyhedra with all vertices on the ideal hyperbolic boundary, for compact polytopes with only finite (or usual) vertices, and for finite volume polyhedra with vertices of both types.
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