Finsleroid-Finsler parallelism. Cosmological aspects

The lucky (and very unique) position of the Finsleroid-Finsler geometry is the occurrenceof a simple algebraic representation for the scalar product, and hence for the angle, of two vectors. In the present paper it is shown that the Finsleroid-type parallel transportation of vectors retains the scalar product and angle unchanged, in the Landsberg-type case. The two-vector extension of the Finsleroid metric tensor is proposed. The Total Category of Parallelism is obtained, in which all the distant-parallelism concepts are meaningful as well as representable in an explicit and simple way. The conclusions made can be reformulated to apply in the relativistic pseudo-Finsleroid framework. Respective gravitational field equations and cosmological metric are discussed.