Infinitesimally natural principal bundles

We extend the notion of a natural fibre bundle by requiring diffeomorphisms of the base to lift to automorphisms of the bundle only infinitesimally, i.e. at the level of the Lie algebra of vector fields. We classify the principal fibre bundles with this property. A version of the main result in this paper (theorem 4.4) can be found in Lecomte's work [12]. Our approach was developed independently, uses the language of Lie algebroids, and can be generalized in several directions.