On a Bipolar Model of Hyperbolic Geometry and its Relation to Hyperbolic   Robertson-Walker Space
  Negatively curved, or hyperbolic, regions of space in an FRW universe are a realistic possibility. These regions might occur in voids where there is no dark matter with only dark energy present. Hyperbolic space is strange and various "models" of hyperbolic space have been introduced, each offering some enlightened view. In the present work we develop a new bipolar model of hyperbolic geometry, closely related to an existing model - the band model - and show that it provides new insights toward an understanding of hyperbolic as well as elliptic Robertson-Walker space and the meaning of its isometries. In particular, we show that the circular geodesics of a hyperbolic Robertson-Walker space can be referenced to two real centers - a Euclidean center and an offset hyperbolic center. These are not the Euclidean center or poles of the bipolar coordinate system but rather refer to two distinct centers for circular orbits of particles in such systems. Considering the physics of elliptic RW space is so well confirmed in the Lambda-CDM model with respect to Euclidean coordinates from a Euclidean center, it is likely that the hyperbolic center plays a physical role in regions of hyperbolic space. 
Harry I. Ringermacher and Lawrence R. Mead
gr-qc astro-ph.CO