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.
Categories: math.DG math.DS