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Let $\Delta$ be a hyperbolic triangle with a fixed area $\varphi$. We prove that for all but countably many $\varphi$, generic choices of $\Delta$ have the property that the group generated by the $\pi$--rotations about the midpoints of the sides of the triangle admits no nontrivial relations. By contrast, we show for all $\varphi\in(0,\pi)\setminus\mathbb{Q}\pi$, a dense set of triangles does afford nontrivial relations, which in the generic case map to hyperbolic translations. To establish this fact, we study the deformation space $\mathfrak{C}_\theta$ of singular hyperbolic metrics on a torus with a single cone point of angle $\theta=2(\pi-\varphi)$, and answer an analogous question for the holonomy map $\rho_\xi$ of such a hyperbolic structure $\xi$. In an appendix by X.~Gao, concrete examples of $\theta$ and $\xi\in\mathfrak{C}_\theta$ are given where the image of each $\rho_\xi$ is finitely presented, non-free and torsion-free; in fact, those images will be isomorphic to the fundamental groups of closed hyperbolic 3--manifolds., Comment: 32 pages. To appear in Math. Z

Falbel, Koseleff and Rouillier computed a large number of boundary unipotent CR representations of fundamental groups of non compact three-manifolds. Those representations are not always discrete. By experimentally computing their limit set, one can determine that those with fractal limit sets are discrete. Many of those discrete representations can be related to (3,3,n) complex hyperbolic triangle groups. By exact computations, we verify the existence of those triangle representations, which have boundary unipotent holonomy. We also show that many representations are redundant: for n fixed, all the (3,3,n) representations encountered are conjugate and only one among them is uniformizable.

We describe a general procedure to produce fundamental domains for complex hyperbolic triangle groups. This allows us to produce new nonarithmetic lattices, bringing the number of known nonarithmetic commensurability classes in PU(2,1) to 22.

The classical Bohr inequality is studied in the Poincare disk model of the hyperbolic plane. A sharp hyperbolic Bohr radius of is obtained for analytic self-maps of the hyperbolic unit disk. The hy...

In this paper, we study the deformation of triangle groups of type ( 3 , n , ∞ ) determined by three R -circles R 0 , R 1 , R 2 with only rotational symmetry, which generalizes the problem studied by Falbel and Parker in [4] .

For a factorized third-order hyperbolic equation on the plane, we obtain sufficient conditions for the solvability of some boundary value problems with conditions that have not been considered for this equation earlier.

In this paper, we obtain upper bounds for the first eigenvalue of the stability operator of a closed constant mean curvature hypersurface Σn immersed either in the Euclidean space Rn+1 or in the hyperbolic space Hn+1, n ≥ 2, in terms of the mean curvature and the length of the total umbilicity operator of Σn. As application, we derive a nonexistence result concerning strong stable hypersurfaces in these ambient spaces. Furthermore, through the calculus of the first stability eigenvalue of circular cylinders in Rn+1 and of hyperbolic cylinders in Hn+1, we present a conjecture related to the first stability eigenvalue of complete constant mean curvature hypersurfaces immersed either in Rn+1 or in Hn+1.Keywords: Euclidean and hyperbolic spaces, closed and complete H-hypersurfaces, first stability eigenvalue, geodesic spheres, circular and hyperbolic cylinders

Inequalities for the lengths of edges in trihedrals of the eep(I) and hhp(I) types on a hyperbolic plane $$\widehat{H}$$ of positive curvature are proved. Based on these inequalities, the extremal property of elliptic and hyperbolic segments and the extremal property of the diagonals in a simple 4-contour on the plane $$\widehat{H}$$ are obtained.

In this paper we investigate the discreteness of complex hyperbolic triangle groups of type (m, n, ∞). For the special case n = ∞, more explicit conclusions are given.

SummaryWe state a formula for the Pythagorean theorem that is valid in Euclidean, spherical, and hyperbolic geometries and give a proof using only properties the geometries have in common.

Preface. How to Use this Book. 0. Historical Strands of Geometry. 1. What is Straight? 2. Straightness on Spheres. 3. What Is an Angle? 4. Straightness on Cylinders and Cones. 5. Straightness on Hyperbolic Planes. 6. Triangles and Congruencies. 7. Area and Holonomy. 8. Parallel Transport. 9. SSS, ASS, SAA, and AAA. 10. Parallel Postulates. 11. Isometries and Patterns. 12. Dissection Theory. 13. Square Roots, Pythagoras and Similar Triangles. 14. Projections of a Sphere onto a Plane. 15. Circles. 16. Inversions in Circles. 17. Projections (Models) of Hyperbolic Planes. 18. Geometric 2-Manifolds. 19. Geometric Solutions of Quadratic and Cubic Equations. 20. Trigonometry and Duality. 21. Mechanisms. 22. 3-Spheres and Hyperbolic 3-Spaces. 23. Polyhedra. 24. 3-Manifolds-the Shape of Space. Appendix A: Euclid's Definitions, Postulates, and Common Notions. Appendix B: Constructions of Hyperbolic Planes. Bibliography. Index.

In this paper we consider ultra-parallel complex hyperbolic triangle groups of type $[m_1,m_2,0]$, i.e. groups of isometries of the complex hyperbolic plane, generated by complex reflections in three ultra-parallel complex geodesics two of which intersect on the boundary. We prove some discreteness and non-discreteness results for these groups and discuss the connection between the discreteness results and ellipticity of certain group elements., 23 pages, 4 figures

In this paper we give lower and upper bounds for the volume growth of a regular hyperbolic simplex, namely for the ratio of the n-dimensional volume of a regular simplex and the \((n-1)\)-dimensional volume of its facets. In addition to the methods of U. Haagerup and M. Munkholm we use a third volume form based on the hyperbolic orthogonal coordinates of a body. In the case of the ideal, regular simplex our upper bound gives the best known upper bound. On the other hand, also in the ideal case our general lower bound, improved the best known one for \(n=3\).

We introduce the notion of “s”-convolution in the hyperbolic plane $$ {\mathrm{\mathbb{H}}}^2 $$ and consider its properties. Analogs of the Helgason spherical transform in the spaces of compactly supported distributions in $$ {\mathrm{\mathbb{H}}}^2 $$ are investigated. We prove a Paley–Wiener–Schwartz-type theorem for the indicated transforms.

In this work ternary non-stationary subdivision schemes, based on hyperbolic B-spline basis functions, have been presented. The proposed hyperbolic, ternary three point and four point subdivision schemes, give pleasing as well as consistent curves with the control polygons as compared to the existing non-stationary subdivision schemes obtained from trigonometric B-splines. The main speciality of the hyperbolic schemes is that, they can reproduce hyperbolas and parabolas quite efficiently which has been demonstrated with the help of curvature plots.

The family of Euclidean triangles having some fixed perimeter and area can be identified with a subset of points on a nonsingular cubic plane curve, i.e., an elliptic curve; furthermore, if the perimeter and the square of the area are rational, then the curve has rational coordinates and those triangles with rational side lengths correspond to rational points on the curve. We first recall this connection, and then we develop hyperbolic analogs. There are interesting relationships between the arithmetic on the elliptic curve (rank and torsion) and the family of triangles living on it. In the hyperbolic setting, the analogous plane curve is a quartic with two singularities at infinity, so the genus is still 1. We can add points geometrically by realizing the quartic as the intersection of two quadric surfaces. This allows us to construct nontrivial examples of rational hyperbolic triangles having the same inradius and perimeter as a given rational right hyperbolic triangle., Comment: 14 pages, 7 figures

We consider the mixed problem for one-dimensional hyperbolic system of thermal conductivity equations. We construct a class of boundary controls that provide given distribution on phase vector (T, q) in a given moment of time. From this class we choose a control by the Lagrange method that minimizes a square functional of loss.

An abelian group of two-dimensional conformal hyperbolic numbers is investigated. A characteristic equation of a hyperbolic number is derived using the theory of permanents. A conformal multiplier, which depends on components of a hyperbolic number or its hyperbolic angle, is defined, and generalized hyperbolic functions are considered. A geometric representation of the group is introduced. It is shown that modulus of a hyperbolic number gives a metric for a two-dimensional Finsler geometry with a quadratic form and a conformal multiplier. A one-dimensional parameter is derived by requiring that the form of the metric be invariant. A group of parameters and functions of hyperbolic numbers are also investigated.

In the 80s of the XXth century the Riemann method for hyperbolic equations of the second order was extended to hyperbolic systems in general form with one spatial variable. In this paper, this result is extended to hyperbolic systems in general form with several spatial variables with time dependent coefficients.

In the present paper from Derevnin–Mednykh’s formula we obtain integral formulas for the volume of an arbitrary hyperbolic octahedron with mmm- and 2|m-symmetry in terms of dihedral angles.

In this paper we study discreteness of complex hyperbolic triangle groups of type $[m,m,0;3,3,2]$, i.e. groups of isometries of the complex hyperbolic plane generated by three complex reflections of orders $3,3,2$ in complex geodesics with pairwise distances $m,m,0$. For fixed $m,$ the parameter space of such groups is of real dimension one. We determine intervals in this parameter space that correspond to discrete and to non-discrete triangle groups., Comment: 18 pages, 3 figures

We study properties of hyperbolic and elliptic cycles of a hyperbolic plane Ĥ of positive curvature. An analog of the Pythagorean theorem for a right triangle with a parabolic hypotenuse is proved. For each type of lines, we obtain formulas expressing the length of a chord of a hyperbolic cycle in terms of the radius of the cycle, the measure of the central angle corresponding to the chord, and the radius of curvature of Ĥ. The plane Ĥ is considered in the projective interpretation. Bibliography: 11 titles.

For a class of second-order partial differential equations of the hyperbolic type with constant coefficients, we study the existence and representation of solutions bounded on the entire plane.

Let M be a quotient of H-2 x ... x H-2 (product of hyperbolic planes) by a uniform lattice of. PSL2(R))(n). We prove that, among metrics of M of prescribed volume, the sum of hyperbolic metrics has minimal volume entropy.

In this study, we consider the kinematics of the hyperbolic plane and define the notions of inflection curve, circling-point curve, cubic of twice stationary curvature curve, and cubic of thrice stationary curve. We also obtain Cartesian and parametric equations of these curves and illustrate some special cases. Finally, we investigate the hyperbolic Ball point, the ordinary and the sixth-order Burmester points in the case of a finite instant pole. https://doi.org/10.1177/1081286515616283

In this paper, we derive the Derevnin–Mednykh integral expression for the volume of a hyperbolic tetrahedron in terms of dihedral angles from the Murakami–Yano expression. As a consequence, we derive an expression for the volume of a hyperbolic tetrahedron in terms of edge lengths.

We study the Dirichlet–Cauchy problem for nonlinear hyperbolic equation of second order in a domain with edges. The aim of this paper is to prove the regularity of solution in weighted Sobolev spaces.

For any three-coloring of the plane, it is shown that there are continuum many triangles of a prescribed type (acute-angled, right-angled, obtuse-angled, isosceles) such that the vertices of each of them carry all three colors.

We give a complete census of orientable cusped hyperbolic 3-manifolds obtained by gluing at most ten regular ideal hyperbolic tetrahedra. Although the census is exhaustive, the question of nonhomeomorphism remains open for some pairs of manifolds with one, two, and three cusps.

The geometry of hyperbolic orbits suggests that Minkowskian geometry, and not Euclidean, may provide the most adequate description of the motion. This idea is explored in order to derive a new regularized formulation for propagating arbitrarily perturbed hyperbolic orbits. The mathematical foundations underlying Minkowski space–time $$\mathbb {M}$$ are exploited to describe hyperbolic orbits. Hypercomplex numbers are introduced to define the rotations, vectors, and metrics in the problem: the evolution of the eccentricity vector is described on the Minkowski plane $$\mathbb {R}_1^2 $$ in terms of hyperbolic numbers, and the orbital plane is described on the inertial reference using quaternions. A set of eight orbital elements is introduced, namely a time-element, the components of the eccentricity vector in $$\mathbb {R}_1^2 $$ , the semimajor axis, and the components of the quaternion defining the orbital plane. The resulting formulation provides a deep insight into the geometry of hyperbolic orbits. The performance of the formulation in long-term propagations is studied. The orbits of four hyperbolic comets are integrated and the accuracy of the solution is compared to other regularized formulations. The resulting formulation improves the stability of the integration process and it is not affected by the perihelion passage. It provides a level of accuracy that may not be reached by the compared formulations, at the cost of increasing the computational time.

We solve an analogue of the Barker–Larman problem for convex polygons in the hyperbolic plane.

The cross ratio is a well known and useful invariant in Euclidean and projective geometries. Since it is invariant under Mobius transformations, it extends to spherical and hyperbolic geometries. I...