On the rectilinear congruences of Lorentz manifold [R 3,(+,+,−)] establishing a mapping between its focal surfaces, preserving the mean curvatures

Abstract

This paper contains four paragraphs. In the first paragraph we consider a hyperbolic rectilinear congruence of the Lorentz manifold [R3,(+,+,−)] the linear element of the system of three partial differential equations in which is reduced the problem of determing the rectilinear congruences of the Lorentz manifold (R3,g) the straight lines of which establish a mapping between their focal surfaces which preserves the mean curvatures (locally). In the second paragraph under the assumptions (2.1) we reduce this system equivalently to the system (2.7) and (2.8) and exspress the theorem 2.1. In the third paragraph under the assumptions (3.1) we succeed to give the solution of the system and exspress the theorem 3.1. In the fourth paragraph we suppose that the linear element of the spherical representation in known (assumption 4.2) and give the class of rectilinear congruences the semidistance of which is given by (4.13) with the mentioned property and express the theorem 4.1.