Finsleroid--Relativistic Space Endowed With Scalar Product
  When a single time-like vector is distinguished geometrically to present the only preferred direction in extending the pseudoeuclidean geometry, the hyperboloid may not be regarded as an exact carrier of the unit-vector image. So under respective conditions one may expect that some time-assymetric figure should be substituted with the hyperboloid. To this end we shall use the pseudo-Finsleroid. The spatial-rotational invariance (the P-parity) is retained. The constant negative curvature is the fundamental property of the pseudo-Finsleroid surface. The present paper develops the approach in the direction of evidencing the concepts of angle, scalar product, and geodesics. In Appendices we shortly outline the basic aspects that stem from the choice of the Finsleroid-relativistic metric functions. 
G.S. Asanov