On canonical Cartan connections associated to filtered G-structures

TL;DR

This paper investigates the existence and explicit construction of canonical Cartan connections for filtered G-structures, generalizing parabolic geometries and providing a Lie algebra-based framework for their analysis.

Contribution

It extends Morimoto's theorem to broader classes of filtered G-structures, allowing for general models and normalization conditions, with an explicit Lie algebra approach.

Findings

Provides a criterion for the existence of canonical Cartan connections.

Offers an explicit Lie algebra-based construction method.

Simplifies verification to finite-dimensional algebraic checks.

Abstract

A filtered manifold is a smooth manifold $M$ together with a filtration of the tangent bundle by smooth subbundles which is compatible with the Lie bracket of vector fields in a certain sense. The Lie bracket of vector fields then induces a bilinear operation on the associated graded of each tangent space of $M$ making it into a nilpotent graded Lie algebra. Assuming that these symbol algebras are the same for all points, one obtains a natural frame bundle for the associated graded to the tangent bundle, and filtered G--structures are defined as reductions of structure group of this bundle. Generalizing the case of parabolic geometries, this article is devoted to the question of whether a filtered G-structure of given type determines a canonical Cartan connection on an extended bundle. As for existence, the result are roughly as general as Morimoto's theorem from 1993, but it has several specific features. First, we allow for general candidates for a homogeneous model and a general version of normalization conditions. Second, the construction is entirely phrased in terms of Lie algebra valued forms and leads to an explicit characterization of the canonical Cartan connection. To verify that the procedure can be applied to a given type of filtered G-structures, only finite dimensional algebraic verifications have to be carried out.