Discrete hyperbolic geometry

The aim of the paper is to make geometers and combinatorialists familiar with old and new connections between the geometry of Lorentz space and combinatorics. Among the topics treated are equiangular lines and their relations to spherical 2-distance sets; spherical and hyperbolic root systems and their relation to graphs whose second largest eigenvalue does not exceed one or two, respectively; and work of Niemeier, Vinberg, Conway and Sloane on Euclidean and Lorentzian unimodular lattices. The first author gratefully acknowledges the support of the Dutch organization for pure research, Z. W. O., during Sept.—Dec. 1980, thus allowing him to spend four months in Eindhoven (Netherlands).     

On the geometry of linear Weingarten hypersurfaces in the hyperbolic space

In this paper, we apply some forms of generalized maximum principles in order to study the geometry of complete linear Weingarten hypersurfaces with nonnegative sectional curvature immersed in the hyperbolic space. In this setting, under the assumption that the mean curvature attains its maximum, we prove that such a hypersurface must be either totally umbilical or isometric to a hyperbolic cylinder.     

Affine geometry over free unitary modules

In this paper we consider systems (P,L,,), where P is an arbitrary non-empty set, L a set of subsets of p and resp. a relation on PxP resp. LxL. In successive stages adding geometrical axioms, we characterize the class of those structures (P,L, ,), which coincides — up to isomorphisms — with the class of all Affine Barbilian-Spaces.     

Integral geometry of unitary area measures

The existence of kinematic formulas for area measures with respect to any connected, closed subgroup of the orthogonal group acting transitively on the unit sphere is established. In particular, the kinematic operator for area measures is shown to have the structure of a co-product. In the case of the unitary group the algebra associated to this co-product is described explicitly in terms of generators and relations. As a consequence, a simple algorithm that yields explicit kinematic formulas for unitary area measures is obtained.     

Ray transforms in hyperbolic geometry

We derive explicit inversion formulae for the attenuated geodesic and horocyclic ray transforms of functions and vector fields on two-dimensional manifolds equipped with the hyperbolic metric. The inversion formulae are based on a suitable complexification of the associated vector fields so as to recast the reconstruction as a Riemann–Hilbert problem. The inversion formulae have a very similar structure to their counterparts in Euclidean geometry and may therefore be amenable to efficient discretizations and numerical inversions. An important field of application is geophysical imaging when absorption effects are accounted for.     

Apollonian packings and hyperbolic geometry

We review the generalized apollonian packings by Bessis and Demko from 3-dimensional viewpoints and solve their conjectures on the discreteness of the groups they constructed. Moreover, we systematically generalize the construction of packings in terms of the Coxeter group theory, and propose a computational algorithm to draw the pictures efficiently based on the automatic group theory.     

Expansive dynamics and hyperbolic geometry

LetM be a compact Riemannian manifold with no conjugate points such that its geodesic flow is expansive. Then we show that the universal Riemannian covering ofM is a hyperbolic geodesic space according to the definition of M. Gromov. This allows us to extend a series of relevant geometric and topological properties of negatively curved manifolds toM and in particular, geometric group theory applies to the fundamental group ofM.     

The hyperbolic geometry of the symmetrized bidisc

We solve the Caratheodory and Kobayashi extremal problems for the open symmetrized bidisc

Universal hyperbolic geometry I: trigonometry

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 paper gives 92 foundational theorems.     

Construction of Cartesian authentication codes from unitary geometry

In the present paper several constructions of Cartesian authentication codes from unitary geometry over finite fields are presented and their size parameters are computed. Assuming that the encoding rules are chosen according to a uniform probability distribution, the probabilities P _I and P _S of a successful impersonation attack and a successful substitution attack, respectively, of these codes are also computed. Moreover, those codes so constructed, for which P _I and P _S are nearly optimal, are also determined.     

Integral geometry of equidistants in hyperbolic space

We generalize the classical formulas of integral geometry, by getting integral geometric formulas for the intersection of a fixed compact hypersurface of hyperbolic space and a moving totally umbilical hypersurface. In particular we compute the mean value of the volume, the total mean curvatures and the Euler characteristic of these intersections when the totally umbilical hypersurface moves over all the intersecting positions. Analogous formulas are given for totally umbilical hypersurfaces contained in totally geodesic planes of ℍⁿ. Work partially supported by MECD grant number EX2003-0987, and MCYT grant number BFM2003-03458.