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A discrete conformality for hyperbolic polyhedral surfaces is introduced in this paper. This discrete conformality is shown to be computable. It is proved that each hyperbolic polyhedral metric on a closed surface is discrete conformal to a…
We study the geodesics of the singularity free metric considered in the preceding Paper I and show that they are complete. This once again demonstrates the absence of singularity. The geodesic completeness is established in general without…
Properties of solutions of generic hyperbolic systems with multiple characteristics with diagonalizable principal part are investigated. Solutions are represented as a Picard series with terms in the form of iterated Fourier integral…
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,…
A negatively curved hyperbolic cone metric is called rigid if it is determined (up to isotopy) by the support of its Liouville current, and flexible otherwise. We provide a complete characterization of rigidity and flexibility, prove that…
Various integrals over elliptic integrals are evaluated as couplings on spheres, resulting in some integral and series representations for the mathematical constants $\pi$, $G$ and $\zeta(3)$.
We show that for given four points on the sphere and prescribed angles at these points, which are not multiples of $2\pi$, the number of metrics of curvature 1 having conic singularities with these angles at these points is finite.
In this paper, we establish a geometric correspondence between constant curvature one metrics with two conical singularities on $S^{2}$ and isometric immersions into Euclidean 3-space $\mathbb{E}^{3}$. Specifically, we explicitly construct…
Let $S$ denote the unit sphere of a real normed space. We show that the intrinsic metric on $S$ is strongly equivalent to the induced metric on $S$. Specifically, for all $x,y\in S$, \[ \|x-y\|\leq d(x,y)\leq\sqrt{2}\pi\|x-y\|, \] where $d$…
In this article, we use the second intrinsic volume to define a metric on the space of homothetic classes of Gaussian bounded convex bodies in a separable real Hilbert space. Using kernels of hyperbolic type, we can deduce that this space…
In the first part of this investigation, [Ha], we generalized a weighted distance function of [Li] and found necessary and sufficient conditions for it being a metric. In this paper some properties of this so-called M-relative metric are…
We prove the isodiametric inequality in the spherical and in the hyperbolic space
The two parameter generalization of the complete elliptic integral of the second kind discussed recently by Barsan is expressed in terms of ordinary complete elliptic integrals.
The space of shapes of a polyhedron with given total angles less than 2\pi at each of its n vertices has a Kaehler metric, locally isometric to complex hyperbolic space CH^{n-3}. The metric is not complete: collisions between vertices take…
A simple proof is given of the necessary and sufficient condition on a triple of positive numbers A,B,C for the existence of a conformal metric of constant positive curvature on the sphere, with three conic singularities of total angles…
We propose a conjecture that the monodromy group of a singular hyperbolic metric on a non-hyperbolic Riemann surface is {\it Zariski dense} in ${\rm PSL}(2,\,{\Bbb R})$. By using meromorphic differentials and affine connections, we obtain…
The density of the hyperbolic metric on the complement of a rectangular lattice is investigated. The question is related to conformal mapping of symmetric circular quadrilaterals with all zero angles.
Let $(M, \partial M)$ be a compact 3-manifold with boundary, which admits a convex co-compact hyperbolic metric. We consider the hyperbolic metrics on $M$ such that the boundary is smooth and strictly convex. We show that the induced…
Dual spherical conchoidal motion has been defined by Yapar. In this work, we define this motion on a dual hyperbolic unit sphere in the dual Lorentzian space with dual signature, and the results carried to the Lorentzian lines space by…
This paper constructs hyperbolic polyhedral metrics via circle packings. We introduce the curvature of circles as a parameter to include all three types of constant curvature curves in the hyperbolic geometry. This provides a unified…