چكيده به لاتين
In this thesis we study the Cartan equivalence problem, focusing on two main methods, namely direct equivalence method and gauge equivalence method.
In chapter 1, some basic introduction of smooth manifolds are presented. Among some definitions we mentioned the definition and elementary theorems of Lie group and Lie algebra.
In chapter 2, the Cartan equivalence problem is investigated and some basic subjects are reviewed.
In chapter 3, two versions of the equivalence problem determining when two second-order differential operators on the line are the same under a change of variables are solved completely using the Cartan method of equivalence.
In chapter 4, we solve the equivalence problem for two third-order differential operators on the line, using the Cartan method of equivalence. Moreover on these differential operators, two versions of the equivalence problems via the direct equivalence problem and gauge equivalence is studied in order to determine conditions such that there exists a fiber-preserving transformation mapping one to the other.
Chapter 4 is devoted to the equivalence problem for fourth-order differential operators with one variable under general fiber-preserving transformation using the Cartan method of equivalence. Similar to chapter 3 two versions of equivalence problems are considered.
Keywords: Lie symmetry, Maurer-Cartan forms, Cartan equivalence problem, Differential operators
