  In a natural way, the local diffeomorphisms of a manifold onto itself act on
the reference frame bundles of any order and on the bundles associated with
them. Due to the transitivity, the invariants by diffeomorphisms of an
associated bundle correspond to the real functions on the orbit space of the
action of the jet group on the typical fiber, which enables to construct the
said associated bundle.
  Scalar differential invariants are functions on bundles of jets of sections
of associated bundles, which are invariant by the standard action of local
diffeomorphisms. We show that these jet bundles are, in turn, associated
bundles with reference frame bundles of higher order. By the previous result,
this will allow to recognize scalar differential invariants as real functions
over a quotient space, regardless of the base manifold.
  We apply the above to $G$-structures, and describe their scalar differential
invariants as functions over certain quotient spaces. Each quotient space
contains a dense differentiable manifold, whose dimension is "the number of
functionally independent scalar differential invariants". We will obtain a
lower bound of that number and calculate it for some cases, in particular for
the scalar conformal differential invariants.
