General Theory of Lie Derivatives

It has been clarified recently that one can define the generalized Lie derivative $${\tilde L_{\left( {\xi ,\eta } \right)}}s$$ of any smooth map f: M → N with respect to a pair of vector field ξ on M and η on N. Given a section s of a vector bundle E → M and a projectable vector field ξ on E over a vector field ξ on M,the second component L η s: M → E of the generalized Lie derivative $${{\tilde{\mathcal{L}}}_{{\left( {\xi ,\eta } \right)}}}f$$ is called the Lie derivative of s with respect to η. We first show how this approach generalizes the classical cases of Lie differentiation. We also present a simple, but useful comparison of the generalized Lie derivative with the absolute derivative with respect to a general connection. Then we prove that every linear natural operator commutes with Lie differentiation. We deduce a similar condition in the non linear case as well. An operator satisfying the latter condition is said to be infinitesimally natural. We prove that every infinitesimally natural operator is natural on the category of oriented m-dimensional manifolds and orientation preserving local diffeomorphisms.