Objective Rates as Covariant Derivatives on the Manifold of Riemannian Metrics

The subject of so-called objective derivatives in Continuum Mechanics has a long 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 \(\textrm{Met}(\mathcal {B})\) $\text{Met}(\mathcal{B})$ 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 “co-rotational type” unnecessary. For an exhaustive list of objective derivatives found in the literature, we exhibit the corresponding covariant derivative on \(\textrm{Met}(\mathcal {B})\) $\text{Met}(\mathcal{B})$ .