On the remarkable properties of the hyperbolic Whitehead link cone-manifold

Denote by W (m; n) the hyperbolic cone-manifold whose underlying space is the 3-sphere and singular geodesics are formed by two components of the Whitehead link with cone angles 2=m and 2=n. The aim of the paper is to establish the Tangent and Sine Rules relating the complex lengthes of the singular geodesics and the cone angles of W (m; n). An explicit upper bound for the real length of the singular geodesic is also given. Keywords: hyperbolic orbifold, hyperbolic cone-manifold, complex length, the Tangent Rule, the Sine Rule 1. Introduction A 3{dimensional hyperbolic cone{manifold is a riemannian 3{dimensional manifold of constant negative sectional curvature with cone-type singularity along simple closed geodesics. To each component of singular set we associate a real number n 1 such that the cone-angle around the component is 2=n: The concept of the hyperbolic cone-manifold generalizes the hyperbolic manifold which appears in the partial case when all cone-angles are 2: T..