Partial AHS-Structures, their Cartan description and partial BGG sequences

TL;DR

This paper extends the concepts of G-structures and Cartan geometries to manifolds with involutive distributions, introducing partial AHS-structures and BGG sequences, thus broadening the scope of geometric analysis on foliated manifolds.

Contribution

It generalizes the construction of Cartan geometries and BGG sequences to partial AHS-structures on manifolds with involutive distributions, expanding their applicability.

Findings

Canonical Cartan geometries extend to partial AHS-structures.

Sequences of differential operators can be constructed intrinsically for these structures.

Under flatness conditions, these sequences resolve sheaves related to leaf spaces.

Abstract

G-structures and Cartan geometries are two major approaches to the description of geometric structures (in the sense of differential geometry) on manifolds of some fixed dimension $n$. We show that both descriptions naturally extend to the setting of manifolds of dimension $\geq n$ which are endowed with a distinguished involutive distribution $F$ of rank $n$. The resulting ``partial'' structures are most naturally interpreted as smooth families of standard G-structures or Cartan geometries on the leaves of the foliation defined by $F$. We prove that for the special class of AHS-structures (also known as $|1|$-graded parabolic geometries) the construction of a canonical Cartan geometry associated to a G-structure extends to this general setting. As an application, we prove that for partial AHS-structures there is an analog of the machinery of BGG sequences. This constructs sequences of differential operators of arbitrarily high order intrinsic to the structures. Under appropriate flatness conditions, these sequence are fine resolutions of sheaves which locally can be realized as pullbacks of sheaves on local leaf spaces for the foliation defined by $F$.