Natural maps on the iterated jet prolongation of a fibred manifold (Q1175920)

Let \(Y\to X\) be a fibred manifold and \(J^ 1J^ 1Y\) its second iterated jet prolongation. The paper proves that all natural transformations \(J^ 1J^ 1Y\to J^ 1J^ 1Y\) dependent on a linear connection on the base manifold \(X\) constitute a 3-parameter family and interprets them geometrically. A special case is constituted by the involution of \(J^ 1J^ 1Y\). The above result is applied to the transformation of a connection of the fibred manifold \(Y\to X\) and a linear connection on the base manifold \(X\) into a connection of the first jet prolongation \(J^ 1Y\to X\). Furthermore, all natural transformations of the second sesquiholonomic and holonomic jet prolongations of \(Y\to X\) into themselves are determined.