(1) Each pair of points can be joined by one and only one straight line segment. (2) Any straight line segment can be indefinitely extended in either direction. (3) There is exactly one circle of any given radius with any given center. (4) All right angles are congruent to one another. (5) If a straight line falling on two straight lines makes the interior angles on the same side less than two ...

We use hyperbolic geometry to construct simply-connected symplectic or complex manifolds with trivial canonical bundle and with no compatible Kähler structure. We start with the desingularisations of the quadric cone in C: the smoothing is a natural S-bundle over H, its holomorphic geometry is determined by the hyperbolic metric; the small-resolution is a natural S-bundle over H with symplectic...

Suppose M is a compact connected odd-dimensional manifold with boundary, whose interior M comes with a complete hyperbolic metric of finite volume. We will show that the L2-topological torsion of M and the L2-analytic torsion of the Riemannian manifold M are equal. In particular, the L2-topological torsion of M is proportional to the hyperbolic volume of M , with a constant of proportionality w...

A hyperbolic algebraic curve is a bounded subset of an algebraic set. We study the function theory and functional analytic aspects of these sets. We show that their function theory can be described by finite codimensional subalgebras of the holomorphic functions on the desingularization. We show that classical analytic techniques, such as interpolation, can be used to answer geometric questions...

1.1. Let X be a complex analytic manifold or an algebraic variety over a perfect field k of dimension d, let xo be a point in X(k), and let g: X→ gl(n, k) be a germ of a matrix valued analytic (respectively, polynomial) function on X in an analytic (respectively, Zariski) neighborhood of xo. Questions regarding the possibility of reduction of such functions to the Jordan form j(x), or to some o...

These pages serve two purposes. First, they are notes to accompany the talk Hyperbolic and projective geometry in constraint programming for CAD by Walter Whiteley at the János Bolyai Conference on Hyperbolic Geometry, 8–12 July 2002, in Budapest, Hungary. Second, they sketch results that will be included in a forthcoming paper that will present the equivalence of the first-order rigidity theor...

We explain how to construct certain potential functions for the hyperbolic structures of a knot complement, which are closely related to the analytic functions on the deformation space of hyperbolic structures. AMS Classification 57M50; 57M25, 57M27

This paper expands the earlier paper [30] and presents foundation for a systematic treatment of three main (elliptic, parabolic and hyperbolic) types of analytic function theory based on the representation theory of SL2(R) group. We describe here geometries of corresponding domains. The principal rôle is played by Clifford algebras of matching types. In this paper we also generalise the Fillmor...