The geometry of the space of Cauchy data of nonlinear PDEs

TL;DR

This paper develops a geometric framework using flag bundles to describe the space of Cauchy data for nonlinear PDEs, generalizing traditional jet bundle approaches and providing new insights into transversality conditions.

Contribution

It introduces a novel geometric approach using flag bundles for higher-order jet bundles, enhancing the understanding of Cauchy data in nonlinear PDEs.

Findings

Generalized jet bundle theory via flag bundles

Established a geometric framework for Cauchy data

Derived transversality conditions in Calculus of Variations

Abstract

First-order jet bundles can be put at the foundations of the modern geometric approach to nonlinear PDEs, since higher-order jet bundles can be seen as constrained iterated jet bundles. The definition of first-order jet bundles can be given in many equivalent ways - for instance, by means of Grassmann bundles. In this paper we generalize it by means of flag bundles, and develop the corresponding theory for higher-oder and infinite-order jet bundles. We show that this is a natural geometric framework for the space of Cauchy data for nonlinear PDEs. As an example, we derive a general notion of transversality conditions in the Calculus of Variations.