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

TL;DR

This paper explores Cartan's 1931 geometric framework, extending the concept of orthogonality based on area considerations within a variational approach to hypersurfaces in differential geometry.

Contribution

It generalizes Cartan's original notion of orthogonality in his area-based geometry using calculus of variation techniques.

Findings

Extended Cartan's orthogonality concept to broader geometric contexts

Connected area-based metrics with variational principles

Provided a new perspective on Cartan's 1931 geometric construction

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.