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.
"Canonical involution on double jet bundles,"
Turkish Journal of Mathematics: Vol. 41:
4, Article 8.
Available at: https://journals.tubitak.gov.tr/math/vol41/iss4/8