In this paper we introduce a family of examples that can be regarded as
spaces of nonpositive curvature, but with the distinct quality that they are
not complete as metric spaces. This amounts to the fact that they are modelled
on a finite von Neumann algebra, and the metrics introduced arise from the
trace of the algebra. In spite of the noncompleteness of these manifolds, their
geometry can be studied from the view-point of metric geometry, and several
techniques derived from the functional analysis are applied to gain insight on
their geodesic structure.