  The subject of so-called objective derivatives in Continuum Mechanics has
along history and has generated varying views concerning their true
mathematical interpretation. Several attempts have been made to provide a
mathematical definition that would at least partially unify the existing
notions. In this paper, we demonstrate that, under natural assumptions, all
objective derivatives correspond to covariant derivatives on the
infinite-dimensional manifold Met(B) of Riemannian metrics on the body.
Furthermore, a natural Leibniz rule enables canonical extensions from covariant
to contravariant tensor fields and vice versa. This makes the sometimes-used
distinction between objective derivatives of ``Lie type'' and ``corotational
type'' unnecessary. For an exhaustive list of objective derivatives found in
the literature, we exhibit the corresponding covariant derivative on Met(B).
