Homogeneous algebraic distributions (Q5940649)

The goal of this paper is to provide characterization of homogeneous algebraic distributions on vector bundles. The classical result states that a vector field \(X\) on \(R^n\) is homogeneous algebraic of degree \(d\) if and only if \([\chi,X_j]=(d-1)X\), where \(\chi\) is the Liouville vector field. In this paper, the authors generalize the above statement to distributions of arbitrary rank. They show that a vertical distribution \(\mathcal{D}\) on a vector bundle \(p\colon E\to M\) locally spanned by vertical vector fields \(X_1,\dots, X_r\) is homogeneous algebraic of degree \(d\) if and only if there exists an \(r\times r\) matrix \(A=[a_{ij}]\) of smooth functions given by \([\chi,X_j]=\sum_{i=1}^r a_{ij}X_i\) such that \(A\) restricted to the zero section of \(E\) is \((d-1)\) times the identity matrix.