Ideas of E. Cartan and S. Lie in modern geometry: $G$-structures and differential equations. Lecture 1

TL;DR

This paper introduces Cartan and Lie's ideas in modern geometry, focusing on the Cartan reduction method using tools like $G$-structures and jets theory to classify differential equations.

Contribution

It explains the Cartan reduction method and its application to classifying differential equations using $G$-structures and principal bundles.

Findings

Classified 3-webs in $ ext{R}^2$ using Cartan's method.

Demonstrated the use of $G$-structures in differential geometry.

Provided a framework for weak classification of differential equations.

Abstract

This is the lecture 1 of a mini-course of 4 lectures. Our purpose of this mini-curse is to explain some ideas of E. Cartan and S. Lie when we study differential geometry, particularly we will to explain the Cartan reduction method. The Cartan reduction method is a technique in Differential Geometry for determining whether two geometrical structure are the same up to a diffeomorphism. This method use new tools of differential geometry as principal bundles, $G$-structures and jets theory. We start with an example of a $G$-structure: the 3-webs in $\mathbb{R}^{2}$. Here we use the Cartan method to classify the differential equations but not to resolve. This is a classification can be a weak classification in the sense of not involving all the structural invariants.