Prolongation on regular infinitesimal flag manifolds

TL;DR

This paper develops a method to transform certain overdetermined PDE systems on regular infinitesimal flag manifolds into linear systems, enabling dimension bounds of solution spaces via representation theory.

Contribution

It introduces a conceptual framework to rewrite semi-linear PDE systems as linear connections on vector bundles over flag manifolds, facilitating solution space analysis.

Findings

Rewrites overdetermined PDE systems as linear connections.

Provides a way to bound solution space dimensions.

Uses representation theory to compute vector bundle ranks.

Abstract

Many interesting geometric structures can be described as regular infinitesimal flag structures, which occur as the underlying structures of parabolic geometries. Among these structures we have for instance conformal structures, contact structures, certain types of generic distributions and partially integrable almost CR-structures of hypersurface type. The aim of this article is to develop for a large class of (semi-)linear overdetermined systems of partial differential equations on regular infinitesimal flag manifolds $M$ a conceptual method to rewrite these systems as systems of the form $\tilde\nabla(\Sigma)+C(\Sigma)=0$, where $\tilde\nabla$ is a linear connection on some vector bundle $V$ over $M$ and $C: V\rightarrow T^*M\otimes V$ is a (vector) bundle map. In particular, if the overdetermined system is linear, $\tilde\nabla+C$ will be a linear connection on $V$ and hence the dimension of its solution space is bounded by the rank of $V$. We will see that the rank of $V$ can be easily computed using representation theory.