  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.
