Élie Cartan's approach to differential geometry emerged as a continuation of Felix Klein's Erlangen program. The general idea of Cartan geometry is to describe curved analogues of Klein geometries, i.e., homogeneous spaces \(G/H\), where \(G\) is a Lie group and \(H\) its closed Lie subgroup. The merit of this approach is that it develops a suitable differential calculus for studying geometric structures through the algebraic apparatus of Lie theory. As a result, Cartan geometry provides a group-theoretic description of many classical geometric structures (Riemannian, conformal, projective to name a few), making central the notion of a symmetry. This talk aims to give a gentle introduction to the modern formulation of Cartan geometry and showcase various applications of the theory.

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