Further Applications

In this chapter we discuss some further geometric problems about different types of natural operators. First we deduce that all natural bilinear operators transforming a vector field and a differential k-form into a differential k-form form a 2-parameter family. This further clarifies the well known relation between Lie derivatives and exterior derivatives of k-forms. From the technical point of view this problem can be considered as a preparatory exercise to the problem of finding all bilinear natural operators of the type of the FrölicherNijenhuis bracket. We deduce that in general case all such operators form a 10-parameter family. Then we prove that there is exactly one natural operator transforming general connections on a fibered manifold Y → M into general connections on its vertical tangent bundle VY → M. Furthermore, starting from some geometric problems in analytical mechanics, we deduce that all first-order natural operators transforming second-order differential equations on a manifold M into general connections on its tangent bundle TM → M form a one parameter family. Further we study the natural transformations of the jet functors. The construction of the bundle of all r-jets between any two manifolds can be interpreted as a functor J r on the product category Mf m × Mf. We deduce that for r ≥ 2 the only natural transformations of J r into itself are the identity and the contraction, while for r = 1 we have a one-parameter family of homotheties. This implies easily that the only natural transformation of the functor of the r-th jet prolongation of fibered manifolds into itself is the identity. For the second iterated jet prolongation J 1(J 1 Y) of a fibered manifold Y we look for an analogy of the canonical involution on the second iterated tangent bundle TTM. We prove that such an exchange map depends on a linear connection on the base manifold and we give a simple list of all natural transformations of this type.