BGG Sequences | A Riemannian perspective

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, occur as 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 $\Bbb R^n$ 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 $\Bbb R^n$. 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.

来自半单李代数（semisimple Lie algebras）表示论的BGG分解（BGG resolutions）与广义BGG分解（generalized BGG resolutions），已被推广至赋予了属于抛物几何（parabolic geometries）族的几何结构的流形上的不变微分算子序列。其中两类结构分别为共形结构（conformal structures）与射影结构（projective structures），二者分别对应黎曼度量（Riemannian metric）与切丛（tangent bundle）上指定无挠联络（torsion-free connection）的弱化形式。特别地，作为该构造的极特殊情形，可在$Bbb R^n$的开子集上得到BGG序列。后续研究表明，这类序列的若干实例在应用数学中颇具研究价值，因其可用于构建数值方法，以研究与弹性力学（elasticity theory）、数值相对论（numerical relativity）及相关领域相关的算子。本文旨在搭建介于抛物几何框架下的BGG序列与$Bbb R^n$区域情形之间的中间理论层级。我们将构造黎曼流形上的共形BGG序列，以及切丛配备保体积线性联络（volume preserving linear connection）的流形上的射影BGG序列。此类构造无需借助抛物几何的相关前置知识，除标准微分几何方法外，仅需表示论作为深层理论支撑。因此，本文所得结论既可视为抛物几何构造的显式简化版本，也可看作将$Bbb R^n$区域的简化构造推广至一般黎曼流形，抑或推广至切丛上配备恰当联络的流形。