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

Andreas Cap, Micha Andrzej Wasilewicz

TL;DR

The paper extends the core frameworks of G-structures and Cartan geometries to partial geometries defined by an involutive distribution $F$, interpreting them as leaf-wise structures along the foliation of $F$. It shows that partial AHS-structures admit a canonical normal partial Cartan description and develops a corresponding partial BGG machinery, yielding higher-order intrinsic differential operators and, under flatness, resolutions of leaf-space sheaves. The construction is functorial and yields strong uniqueness results, linking partial Cartan geometries to underlying partial $G_0$-structures and enabling a rich algebraic toolkit via Kostant's theory to produce geometric BGG sequences. This framework broadens the applicability of parabolic-geometric methods to foliated or fibered settings, with potential implications for Legendrean and conformal-type partial geometries and their differential invariants.

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 $F$.

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

TL;DR

The paper extends the core frameworks of G-structures and Cartan geometries to partial geometries defined by an involutive distribution , interpreting them as leaf-wise structures along the foliation of . It shows that partial AHS-structures admit a canonical normal partial Cartan description and develops a corresponding partial BGG machinery, yielding higher-order intrinsic differential operators and, under flatness, resolutions of leaf-space sheaves. The construction is functorial and yields strong uniqueness results, linking partial Cartan geometries to underlying partial -structures and enabling a rich algebraic toolkit via Kostant's theory to produce geometric BGG sequences. This framework broadens the applicability of parabolic-geometric methods to foliated or fibered settings, with potential implications for Legendrean and conformal-type partial geometries and their differential invariants.

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 . We show that both descriptions naturally extend to the setting of manifolds of dimension which are endowed with a distinguished involutive distribution of rank . 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 . We prove that for the special class of AHS-structures (also known as -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 .

Paper Structure

This paper contains 17 sections, 12 theorems, 26 equations.

Table of Contents

  1. Introduction
  2. Partial AHS-structures and their Cartan description
  3. $G_0$-structures and AHS-structures
  4. Partial $G_0$-structures
  5. Examples
  6. Cartan description of AHS-structures
  7. Partial Cartan geometries
  8. Curvature and normality
  9. Existence of normal partial Cartan geometries
  10. Uniqueness
  11. The partial BGG machinery
  12. The partial fundamental derivative
  13. Partial tractor connection and partial twisted exterior derivative
  14. The algebraic Hodge theory
  15. Converting to geometry
  16. ...and 2 more sections

Key Result

Proposition 4

Let $G_0$ be a covering of a closed subgroup $\underline{G}_0\subset GL(n,\mathbb R)$ and let $K$ be the kernel of the covering homomorphism $G_0\to\underline{G}_0$. (1) For a manifold $M$ of dimension $\geq n$ and an involutive distribution $F\subset TM$, a partial $G_0$-structure on $(M,F)$ is equ

Theorems & Definitions (30)

  • Definition 1
  • Definition 2
  • Definition 3
  • Proposition 4
  • proof
  • Theorem 5
  • Definition 6
  • Proposition 7
  • proof
  • Remark 8
  • ...and 20 more