Double structures and jets

TL;DR

This paper explores the geometric structures of double velocities and contact elements, revealing how their subgroup structures influence the formation of affine bundles with well-defined curvature components.

Contribution

It introduces a new perspective on the double vector bundle structure of double velocities and their relation to holonomic and semiholonomic subgroups in jet group prolongations.

Findings

Identification of holonomic and semiholonomic submanifolds in double contact elements

Construction of affine bundles with curvature components

Mirror structure between double velocities and jet group prolongations

Abstract

We show how the double vector bundle structure of the manifold of double velocities, with its submanifolds of holonomic and semiholonomic double velocities, is mirrored by a structure of holonomic and semiholonomic subgroups in the principal prolongation of the first jet group. We use the actions of these groups to construct holonomic and semiholonomic submanifolds in the manifold of double contact elements, and show that these give rise to affine bundles where a semiholonomic element has well-defined holonomic and curvature components.