On The Equivalence Problem for Geometric Structures, I

TL;DR

This paper investigates the local and global equivalence problems for geometric structures of arbitrary order, emphasizing transformations within pseudo-groups and utilizing PDE frameworks, inspired by Cartan's methods.

Contribution

It introduces a comprehensive approach to the equivalence problem using prolongation spaces and PDE solutions, highlighting Cartan's transformation techniques.

Findings

Framework for analyzing geometric structure equivalence

Application of PDE methods to equivalence problems

Emphasis on Cartan's transformation techniques

Abstract

We discuss the local and global problems for the equivalence of geometric structures of an arbitrary order and, in later sections, attention is given to what really matters, namely the equivalence with respect to transformations belonging to a given pseudo-group of transformations. We first give attention to general prolongation spaces and thereafter insert the structures in their most appropriate ambient namely, as specific solutions of partial differential equations where the equivalence problem is then discussed. In the second part, we discuss applications of all this abstract nonsense and take considerable advantage in exploring \'Elie Cartan's magical trump called transformations et prolongements m\'eri\'edriques that somehow seem absent in present day geometry.