The concept of orthogonality in Cartan's geometry based on the concept of area

Abstract

In 1931 Elie Cartan constructed a geometry which was rarely considered. Cartan proposed a way to define an infinitesimal metric $ds$ starting from a variational problem on hypersurfaces in an $n$-dimensional manifold $\mathcal{M}$. This distance depends not only of the point $\textsc{m}\in\mathcal{M}$ but on the orientation of a hyperplane in the tangent space $T_{\textsc{m}}\mathcal{M}$. His first step is a natural definition of the orthogonal direction to such tangent hyperplane. In this paper we extend it, starting form considerations from the calculus of variation.