Prehomogeneous geometries vs. bundles with connections

TL;DR

This paper compares prehomogeneous geometries and bundles with connections across fibered manifolds, jet bundles, and nonlinear PDEs, highlighting their conceptual relationships and differences.

Contribution

It provides a detailed analysis contrasting prehomogeneous geometries with bundles with connections in various geometric contexts.

Findings

Clarifies the relationship between prehomogeneous geometries and bundles with connections.

Analyzes their roles in fibered manifolds, jet bundles, and nonlinear PDEs.

Highlights differences and similarities in geometric structures.

Abstract

The below discussion is in three sections A, B, C, each section in two parts I, II, I representing the standpoint of bundles with connections and II representing the standpoint of prehomogeneous geometries (phg's). In A, our object of study is a fibered manifold, in B its k-jet bundle and in C a general nonlinear PDE.