Generalized Euler vector fields associated to the Weil bundles (Q2831618)

The Euler vector field on a vector bundle \(E\) is the infinitesimal generator of the \(\mathbb R\)-action on \(E\) by fiber-wise homotheties. Given a manifold \(M\) and a Weil algebra \(A\), the additive \(\mathbb R\) acts on the Weil bundle \(T^A M\) by fiber-wise diffeomorphisms. The infinitesimal generator of this action is, in the author's words, the \textit{generalized Euler vector field on \(T^A M\)}. The generalized Euler vector field \(\xi_{T^A M}\) is natural in a suitable sense and this suggests the author to give the following definition: A \textit{natural Euler vector field associated to \(T^A\)} is a natural transformation (of functors) \(T^A \to T \circ T^A\). NEWLINENEWLINEIn the first part of the paper, the author proves that natural Euler vector fields are in one-to-one correspondence with derivations of \(A\). Later he specializes to some specific \(A\). In the second part of the paper, he defines \textit{homogeneous tensor fields on the Weil bundle}: A tensor field \(\theta\) on \(T^A M\) is homogeneous of degree \(k\) if \(\mathcal L_{\xi_{T^A M}} \theta = k \theta\). Later he studies the behaviour of various tensorial operations (contractions, tensor products, etc.) with respect to the degree. In the last part of the paper, the author defines homogeneous (of degree \(1\)) Dirac structures with respect to a given vector field, and proves that the natural lift of a Dirac structure on \(M\) to the higher tangent bundle \(T^r M\) (which is a special case of Weil bundle) is homogeneous with respect to to \(\frac{1}{r} \xi_{T^r M}\).