In this paper we deal with Voronoi diagram in simply connected complete manifold with non positive curvature, called Hadamard manifold. We prove that a part of the Voronoi diagram can be characterized by hyperbolic Voronoi diagram. Voronoi diagram in simply connected complete manifold is also characterized for a given set of points satisfying a distance condition.

The Voronoi diagram is the most fundamental and useful concept in computational geometry. To understand impacts of non-Euclidean geometry on computational geometry, this paper investigates the Voronoi diagram in hyperbolic space. We first present characterizations of this diagram by means of the Enclidean Voronoi diagram, and based on them propose efficient algorithms to construct it. Some applications are also mentioned.