A p-Sum Representation of the 1-Jets of p-Velocities

Abstract

The theory of jets provides a useful tool for various fields in mathematics, enablingthe solution of higher-order differential equations and partial differential equations that modelcomplex mechanical systems. This theory adopts a geometric approach to generalized mechanicsand field theory. For instance, in Lagrangian particle mechanics, the formalism of higher-order jetbundles proves useful. Thus, the study of jets is not only beneficial to mathematics but also extends itsapplicability to other fields such as physics. In this study, we approach jet bundles from a differentialgeometry perspective. Specifically, we use structure of the bundle of all 1-jets of maps fromRptoMwith source at 0. By employing normal coordinates on the manifoldM, we demonstrate that thisbundle is diffeomorphic to thep-Whitney sum of tangent bundles. Then, we prove that this bundlecarries a vector bundle structure. Using its vector bundle structure, the paper establishes the existenceof the isomorphism for tangent bundles ofp1velocities, and extends the previous result by provingthat the vector bundle of 1-jets ofp-velocities is isometric top-sum of tangent bundles, even in caseswhere the base manifold does not carry a Banach structure.