Differential geometry, Jet spaces, Differential equations, Cartan distribution, Contact transformation, Hyperbolic equations, Darboux method, Laplace invariants, Laplace transformations.

In this master thesis we will consider the hyperbolic equations of the form F(x,y,u,ux,uy,uxx,uxy,uyy) = 0, with the aim of studying the geometric aspects of Laplace and Darboux methods. In Chapter 1 we will start with the basic of classical theory of Laplace transformations and Laplace invariants for linear hyperbolic equations. In particular, in the first two sections 1.1-1.2 of that Chapter, we will present the necessary preliminaries on the line congruences used to construct Laplace transformations of surfaces in R3. Subsequently, by minig the ideas behind Laplace transformations of surfaces, in Section 1.3 we will introduce the Laplace transformations and Laplace invariants for linear hyperbolic equations.Then, coming to the last two sections, in Section 1.4 we will discuss the equivalence problem of linear hyperbolic equations, whereas Section 1.5 we will discuss some properties of the equations that are preriodic with respect to Laplace transformations. Later, in Chapter 2, we will approach the hyperbolic equations as submanifolds of jet spaces. In Section 2.1 we will provide a short review of the main geometric aspects of the theory of jet bundle.Then in Section 2.2 we will discuss the notion of characteristics for second-order partial differential equations, whereas in Section 2.3 we will study the fundamental properties of characteristic vector fields on the infinite prolongation of a second-order equation. Subsequently, in Section 2.4, we will introduce the notion of Darboux integrability together with some examples that provie a first insight to the Darboux integration method. Later in Section 2.5, we will present the universal linearization and its equivalent forms in terms of the projected Lie derivatives with respect to the characteristic vector fields, together with the definition of generalized Laplace invariants for an hyperbolic equation. Finally, in Section 2.6 we will study a criterion for Darboux integrability, in terms of generalized Laplace invariants.