The main objective of this thesis is to present a geometry given by Cartan in 1933 \cite{Cartan1933}. The Finsler geometry has many analogies with this theory. We studied the outline of this geometry. The starting point of Cartan which is similar to that which leads to the Finsler geometry, is to imagine the space to be made of ''contact elements'', an element being given by a point $M$ in $\mathcal{M}$ and an oriented hyperplan passing through this point in the tangent space $T_M\mathcal{M}^n$. Thus we have defined \textit{Cartan geometry based on the concept of area}. In a first step, I was interested in the notion of orthogonality in this geometry. Cartan's method to state the equivalence problem is a crucial tool. After, we applied this method to Monge-Ampère equations (elliptic case). Hence following the work of R. Bryant, D. Grossman and P. Griffiths in the years 2002-2005 in order to clarify the strategy of Cartan. However, many points have to be explored in order to have a clear dictionary between a modern language as the one used by Bryant, Griffiths and Grossman and that of Cartan.
