BGG Sequences -- A Riemannian perspective

TL;DR

This paper develops simplified constructions of BGG sequences on Riemannian and related manifolds, bridging the gap between abstract parabolic geometry and practical applications in numerical methods for elasticity and relativity.

Contribution

It introduces explicit, representation-theory-based methods for constructing BGG sequences on Riemannian and volume-preserving connection manifolds without relying on full parabolic geometry frameworks.

Findings

Constructed conformal BGG sequences on Riemannian manifolds.

Developed projective BGG sequences on manifolds with volume-preserving connections.

Provided a bridge between abstract theory and practical applications in numerical analysis.

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.