the only justification for the einstein velocity addition law appeared to be its empirical adequacy, so that the intrinsic beauty and harmony in einstein addition remained for a long timea mystery to be conquered. accordingly, the aim of this expository article is to present(i) the einstein relativistic vector addition,(ii) the resulting einstein scalar multiplication,(iii) the einstein relativ...

I. BASIC TOOL: RATIONAL PARAMETRIZATION Algebraic geometry is a generalization of analytic geometry the familiar study of lines, planes, circles, parabolas, ellipses, hyperbolas, and their 3 dimensional versions: spheres, cones, hyperboloids, ellipsoids, and hyperbolic surfaces. The essential common property these all have is that they are defined by polynomials. This is the defining characteri...

Let f be a function that is analytic in the unit disc. We give new estimates, and new proofs of existing estimates, of the Euclidean length of the image under f of a radial segment in the unit disc. Our methods are based on the hyperbolic geometry of plane domains, and we address some new questions that follow naturally from this approach.

The end of a hyperbolic surface is studied in terms of the behavior at innnity of geodesics on the surface. For a class of surfaces called untwisted utes it is possible to give a fairly precise description of the ending geometry. From the point of view of a Fuchsian group representing such a surface this provides new information about the existence of Dirichlet and Garnett points. 0. Introducti...

In this chapter we discuss the theory of complex projective structures on compact surfaces and its connections with Teichmüller theory, 2-and 3-dimensional hyperbolic geometry, and representations of surface groups into PSL 2 (C). Roughly speaking, a complex projective structure is a type of 2-dimensional geometry in which Möbius transformations play the role of geometric congru-ences (this is ...

We give a new real-analytic embedding of the Teichmüller space of closed Riemann surfaces of genus 2 into R6. The parameters are explicitly defined in terms of the underlying hyperbolic geometry. The embedding is accomplished by writing down four matrices in PSL(2, R), where the entries in these matrices are explicit algebraic functions of the parameters. Explicit inequalities are given to defi...

We present a geometric proof of the Poincaré-Dulac Normalization Theorem for analytic vector fields with singularities of Poincaré type. Our approach allows us to relate the size of the convergence domain of the linearizing transformation to the geometry of the complex foliation associated to the vector field. A similar construction is considered in the case of linearization of maps in a neighb...

With his hyperbolic Dehn surgery theorem and later the orbifold theorem, Thurston demonstrated the power of using hyperbolic cone-manifolds to understand complete, non-singular hyperbolic 3-manifolds. Hodgson and Kerckhoff introduced analytic techniques to the study of cone-manifolds that they have used to prove deep results about finite volume hyperbolic 3-manifolds. In this paper we use Hodgs...

We study the symplectic geometry of the moduli spaces Mr = Mr(H ) of closed n-gons with fixed side-lengths in hyperbolic three-space. We prove that these moduli spaces have almost canonical symplectic structures. They are the symplectic quotients of B by the dressing action of SU(2) (here B is the standard Borel subgroup of SL2(C)). We show that the hyperbolic Gauss map sets up a real analytic ...