Canonical Involution on Double Jet Bundles

TL;DR

This paper extends the concept of double tangent bundles to double jet bundles, demonstrating their structures, relationships, and a natural involution interchanging primary and secondary vector bundle structures.

Contribution

It introduces the double jet bundle as a generalization of tangent bundles, establishes their structures, and identifies a natural involution between primary and secondary structures.

Findings

Double jet bundles have two compatible vector bundle structures.

Manifold charts from primary and secondary structures are compatible.

A natural involution interchanges the primary and secondary structures.

Abstract

In this study, we generalize double tangent bundles to double jet bundles. We present a secondary vector bundle structure on a 1-jet of a vector bundle. We show that 1-jet of a vector bundle carries two vector bundle structures, namely primary and secondary structures. We also show that the manifold charts induced by primary and secondary structures belong to the same atlas. We prove that double jet bundles can be considered as a quotient of second order jet bundle. We show that there exists a natural involution that interchanges between primary and secondary vector bundle structures on double jet bundles.