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BGG Sequences -- A Riemannian perspective

Andreas Cap

TL;DR

This work presents a tractor-free, Riemannian- and volume-preserving-connection framework to realize BGG sequences on manifolds, bridging parabolic-geometry theory with explicit constructions on domains in $\mathbb{R}^n$. It builds conformal BGG sequences via a twisted de Rham complex and Kostant-type cohomology, introducing splitting operators to define higher-order BGG operators and clarifying when the resulting sequences form complexes (notably in the conformally flat case). The paper also extends the same philosophy to projective geometry, yielding projective BGG sequences and emphasizing how curvature (Weyl, Cotton–York) governs the (non)complex nature of these sequences, with concrete examples like conformal Killing operators and elasticity-type complexes. Overall, it provides practical, representation-theoretic tools to construct and analyze invariant differential operators on broad geometric backgrounds without full parabolic machinery, highlighting potential numerical and analytical applications in elasticity, relativity, and related fields.

Abstract

BGG resolutions and generalized BGG resolutions from representation theory of semisimple Lie algebras have been generalized to sequences of invariant differential operators on manifolds endowed with a geometric structure belonging to the family of parabolic geometries. Two of these structures, conformal structures and projective structures, occur as weakenings of a Riemannian metric respectively of a specified torsion-free connection on the tangent bundle. In particular, one obtains BGG sequences on open subsets of $\mathbb R^n$ as very special cases of the construction. It turned out that several examples of the latter sequences are of interest in applied mathematics, since they can be used to construct numerical methods to study operators relevant for elasticity theory, numerical relativity and related fields. This article is intended to provide an intermediate level between BGG sequences for parabolic geometries and the case of domains in $\mathbb R^n$. We provide a construction of conformal BGG sequences on Riemannian manifolds and of projective BGG sequences on manifolds endowed with a volume preserving linear connection on their tangent bundle. These constructions do not need any input from parabolic geometries. Except from standard differential geometry methods the only deeper input comes from representation theory. So one can either view the results as a simplified version of the constructions for parabolic geometries in an explicit form. Alternatively, one can view them as providing an extension of the simplified constructions for domains in $\Bbb R^n$ to general Riemannian manifolds or to manifolds endowed with an appropriate connection on the tangent bundle.

BGG Sequences -- A Riemannian perspective

TL;DR

This work presents a tractor-free, Riemannian- and volume-preserving-connection framework to realize BGG sequences on manifolds, bridging parabolic-geometry theory with explicit constructions on domains in . It builds conformal BGG sequences via a twisted de Rham complex and Kostant-type cohomology, introducing splitting operators to define higher-order BGG operators and clarifying when the resulting sequences form complexes (notably in the conformally flat case). The paper also extends the same philosophy to projective geometry, yielding projective BGG sequences and emphasizing how curvature (Weyl, Cotton–York) governs the (non)complex nature of these sequences, with concrete examples like conformal Killing operators and elasticity-type complexes. Overall, it provides practical, representation-theoretic tools to construct and analyze invariant differential operators on broad geometric backgrounds without full parabolic machinery, highlighting potential numerical and analytical applications in elasticity, relativity, and related fields.

Abstract

BGG resolutions and generalized BGG resolutions from representation theory of semisimple Lie algebras have been generalized to sequences of invariant differential operators on manifolds endowed with a geometric structure belonging to the family of parabolic geometries. Two of these structures, conformal structures and projective structures, occur as weakenings of a Riemannian metric respectively of a specified torsion-free connection on the tangent bundle. In particular, one obtains BGG sequences on open subsets of as very special cases of the construction. It turned out that several examples of the latter sequences are of interest in applied mathematics, since they can be used to construct numerical methods to study operators relevant for elasticity theory, numerical relativity and related fields. This article is intended to provide an intermediate level between BGG sequences for parabolic geometries and the case of domains in . We provide a construction of conformal BGG sequences on Riemannian manifolds and of projective BGG sequences on manifolds endowed with a volume preserving linear connection on their tangent bundle. These constructions do not need any input from parabolic geometries. Except from standard differential geometry methods the only deeper input comes from representation theory. So one can either view the results as a simplified version of the constructions for parabolic geometries in an explicit form. Alternatively, one can view them as providing an extension of the simplified constructions for domains in to general Riemannian manifolds or to manifolds endowed with an appropriate connection on the tangent bundle.

Paper Structure

This paper contains 20 sections, 7 theorems, 46 equations.

Table of Contents

  1. Introduction
  2. Conformal BGG sequences on Riemannian manifolds
  3. Background from Riemannian geometry
  4. Algebraic setup for conformal BGG's
  5. Passing to geometry
  6. Examples
  7. The twisted connection
  8. The twisted de Rham sequence
  9. The cohomology bundles
  10. The Riemannian BGG construction
  11. The conformally flat case
  12. The conformally non-flat case
  13. Example
  14. Some general results
  15. Projective BGG sequences
  16. ...and 5 more sections

Key Result

Theorem 2.2

For $\xi,\eta\in\mathfrak X(M)$ and $s\in\Gamma(\mathcal{V}M)$, the curvature $R^{\mathcal{V}}$ of $\nabla^{\mathcal{V}}$ is given by where $W$ and $Y$ are the Weyl curvature and the Cotton--York tensor of $g$, respectively. In particular, the connection $\nabla^{\mathcal{V}}$ is flat if and only if the metric $g$ is conformally flat.

Theorems & Definitions (17)

  • Definition 2.1
  • Theorem 2.2
  • proof
  • Proposition 2.3
  • proof
  • Remark 2.4
  • Definition 2.5
  • Proposition 2.6
  • proof
  • Theorem 2.7
  • ...and 7 more