Semiholonomic jets and induced modules in Cartan geometry calculus

TL;DR

This paper surveys the algebraic structures underlying Cartan geometry, focusing on jets of sections, induced modules, and invariant differential operators, bridging flat and curved geometries in a unified algebraic framework.

Contribution

It elucidates the connection between jets of sections and induced modules in Cartan geometry, extending algebraic tools from flat to curved cases.

Findings

Clarifies the link between jets and induced modules in Cartan geometry

Extends algebraic methods to curved geometries

Provides a comprehensive overview of invariant differential operators

Abstract

The famous Erlangen Programme was coined by Felix Klein in 1872 as an algebraic approach allowing to incorporate fixed symmetry groups as the core ingredient for geometric analysis, seeing the chosen symmetries as intrinsic invariance of all objects and tools. This idea was broadened essentially by Elie Cartan in the beginning of the last century, and we may consider (curved) geometries as modelled over certain (flat) Klein's models. The aim of this short survey is to explain carefully the basic concepts and algebraic tools built over several recent decades. We focus on the direct link between the jets of sections of homogeneous bundles and the associated induced modules, allowing us to understand the overall structure of invariant linear differential operators in purely algebraic terms. This allows us to extend essential parts of the concepts and procedures to the curved cases.