On the genesis of the Cartan-Kähler theory

Abstract

The theory of exterior differential systems plays a crucial role in Cartan's whole mathematical production. As he once recognized, all the germs of his subsequent work were contained there. Indeed, it provided him with powerful technical tools that turned out to be very useful in many different fields such as the theory of partial differential equations, the theory of infinite dimensional Lie groups (Lie pseudogroups) and differential geometry. Nevertheless, scarce attention has been paid to this area of historical research thus far. Although authoritative scholars have investigated the foundation of exterior differential calculus in Cartan's early papers, no specific analysis of Cartan's subsequent works laying the foundations of what nowadays is known as the Cartan-K\ue4hler theory has been yet provided. This article represents a first attempt to remedy this unsatisfactory state of affairs by focusing on Cartan's work on Pfaffian systems at the very beginning of the past century. The analysis of Cartan's relevant papers is preceded by a description of the historical context in which such contributions were conceived. In this respect, special emphasis will be put on some works by Engel and von Weber on Pfaffian systems, which laid the basis for the subsequent geometrical developments of Cartan's theory of exterior differential systems