Abstract
The aim of this paper is to describe the infinitesimal deformation (M, V) of a complex Finsler space family {(M, Lt)}t∈ℝ and to study some of its geometrical objects (metric tensor, non-linear connection, etc). In this circumstances the induced non-linear connection on (M, V) is defined. Moreover we have elaborate the inverse problem, the problem of the first order deformation of the metric. A special part is devoted to the study of particular cases of the perturbed metric.