Deformation of a kinematic surface in a hyperbolic space

We give a representation of the variational problem on a time (space) like surface immersed in a hyperbolic space. The geometric properties of the deformed surfaces are given. The variational problem for the Klein images of 2-parametric continuous motion of a line (kinematic surface) are introduced. The theory of Klein images is applied to a time like and space like congruence. Finally, additional results are mentioned in the concluding remarks. The technique adapted here is based on Cartanʹs methods of moving frames, exterior differential forms and the group of motions [Result. Math. 39 (2001) 1; Tensor. N.S. 51 (3) (1992) 224; Tensor. N.S. 59 (1998) 36; Collect. Math. 45 (1994) 153].