Overdetermined systems, conformal geometry, and the BGG complex

TL;DR

This paper explores the prolongation procedure for overdetermined systems in conformal geometry, based on BGG sequences, and discusses its generalizations to various geometric structures.

Contribution

It presents a Riemannian prolongation procedure derived from BGG sequences, extending the method to conformal and other geometric structures.

Findings

Prolongation procedure simplifies overdetermined systems in conformal geometry.

Construction of invariant differential operators is generalized to various geometric structures.

The approach unifies methods for different geometric settings.

Abstract

This is an expanded version of a series of two lectures given at the IMA summer program "Symmetries and Overdetermined Systems of Partial Differential Equations". The main part of the article describes the Riemannian version of the prolongation procedure for certain overdetermined system obtained recently in joint work with T.P. Branson, M.G. Eastwood, and A.R. Gover. First a simple special case is discussed, then the (Riemannian) procedure is described in general. The prolongation procedure was derived from a simplification of the construction of Bernstein-Gelfand-Gelfand (BGG) sequences of invariant differential operators for certain geometric structures. The version of this construction for conformal structures is described next. Finally, we discuss generalizations of both the prolongation procedure and the construction of invariant operators to other geometric structures.