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, enabling the solution of higher-order differential equations and partial differential equations that model complex mechanical systems. This theory adopts a geometric approach to generalized mechanics and field theory. For instance, in Lagrangian particle mechanics, the formalism of higher-order jet bundles proves useful. Thus, the study of jets is not only beneficial to mathematics but also extends its applicability to other fields such as physics.

In this study, we approach jet bundles from a differential geometry perspective. Specifically, we use structure of the bundle of all 1-jets of maps from $\mathbb{R}^p$ to $M$ with source at 0. By employing normal coordinates on the manifold M, we demonstrate that this bundle is diffeomorphic to the p-Whitney sum of tangent bundles. Then we prove that this bundle carries a vector bundle structure. Using its vector bundle structure, the paper establishes the existence of the isomorphism for tangent bundles of $p^1$ velocities, and extends the previous result by proving that the vector bundle of 1-jets of p-velocities is isometric to p-sum of tangent bundles, even in cases where the base manifold does not carry a Banach structure.