Invariants of Lagrangians on Weil bundles and their classifications
Let \({\mathcal Q}: B \to A\) be an algebra epimorphism of Weil algebras. For maps \(f:\mathbb{R}^k \to M\) to a manifold \(M\), an equivalence relation can be introduced in a natural way to ensure that the equivalence classes \(j^A f\), called \(A\)-velocities, fill in a bundle \(T^A M\) over \(M\), called the Weil bundle [for a good review on Weil bundles see \textit{I. Kolář}, Proc. 24th nat. conf. geom. topol., 127-136 (1996; Zbl 0853.58006)]. Let \({\mathcal Q}_M:T^B M \to T^A M\) denote the canonical extension of \({\mathcal Q}\) from the bundle of \(B\)-velocities to the bundle of \(A\)-velocities over a manifold \(M\). The paper under review studies natural operators transforming either real-valued functions or 1-forms on \(T^A M\) into real-valued functions or 1-forms on \(T^B M\) and proposes their classification when the dimension of \(M\) is sufficiently large. It follows by earlier work of the author, and is in the general setting of the theory of natural transformations in the sense of \textit{I. Kolář, P. Michor}, and \textit{J. Slovák}, [`Natural operations in differential geometry', Springer-Verlag, Berlin (1993; Zbl 0782.53013)].
scientific article; zbMATH DE number 1059630