In this paper, by using Lorentzian matrix multiplication, [Formula: see text]-Tangent operator is obtained in Lorentzian space. The [Formula: see text]-Tangent operators related with planar, spherical and spatial motion are computed via special matrix groups. [Formula: see text]-Tangent operators are related to vectors. Some illustrative examples for applications of [Formula: see text]-Tangent operators are also presented.

In this paper, we give how to define the basic concepts of differential geometry on Dual space. For this, dual tangent vectors that have p as dual point of application are defined. Then, the dual analytic functions defined by Dimentberg are examined in detail, and by using the derivative of the these functions, dual directional derivatives and dual tangent maps are introduced.

Topology studies the properties of spaces that are invariant under any con-tinuous deformation. Topology is needed to examine the properties of the space. Funda-mentally, the most basic structure required to do math in the space is topology. There exists little information on the expression of the basis and topology on dual space. The main point of the research is to explain how to define the basis and topology on dual space Dⁿ. Then, we will study the geometric constructions corresponding to the open balls in D and D², respectively.

In this paper, the analyticity conditions of dual functions are clearly examined and the properties of the concept derivative are given in detail. Then, using the dual order relation, the dual analytic regions of dual analytic functions are constructed such that a collection of these regions forms a basis on $D^n$. Finally, the equivalent of the inverse function theorem in dual space is given by a theorem and proved.