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This is the lecture 1 of a mini-course of 4 lectures. Our purpose of thismini-curse is to explain some ideas of E. Cartan and S. Lie when we studydifferential geometry, particularly we will to explain the Cartan reductionmethod. The Cartan reduction method is a technique in Differential Geometry fordetermining whether two geometrical structure are the same up to adiffeomorphism. This method use new tools of differential geometry as principalbundles, $G$-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 toclassify the differential equations but not to resolve. This is aclassification can be a weak classification in the sense of not involving allthe structural invariants.

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