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 the 1-jet of a vector bundle carries two vector bundle structures, namely primary and secondary structures. | Turk J Math (2017) 41: 854 – 868 ¨ ITAK ˙ c TUB ⃝ Turkish Journal of Mathematics doi: Research Article Canonical involution on double jet bundles 1 1,2,∗ ˘ H¨ ulya KADIOGLU ˙ Department of Primary School Mathematics Teaching, Yıldız Technical University, Esenler, Istanbul, Turkey 2 Department of Mathematics, Idaho State University, Pocatello, Idaho, USA Received: • Accepted/Published Online: • Final Version: 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 the 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 the 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. Key words: Double jet bundle, double vector bundle, second order jets, canonical involution, tangent bundle of higher order 1. Introduction In general, there are two ways to define k -jets. The first definition is based on using the sections of a fibered manifold. In this definition, a k -jet is an equivalence class determined by an equivalence relation ∽k . Two sections of a fibered manifold are called k -related by the relation ∽k if they have the same Taylor polynomial expansion at the point x truncated at order k. This definition usually leads to a geometric approach, which is applied to the study of systems of differential equations (we refer the reader to [1, 2, 4, 9–11] for more details). The second definition of jet bundles is based on using the functions from N to M , where N and M are smooth manifolds. In this definition, a k -jet .
