Objective rates as covariant derivatives on the manifold of Riemannian metrics

Abstract

The subject of so-called objective derivatives (in Continuum Mechanics) has a long history and is somehow controversial. Several works concern the formulation of the correct mathematical definition of what they are really and try to unify them all into one definition (or one family). In this paper, we show, finally, that all of them correspond in fact to covariant derivatives on the infinite dimensional manifold Met(B) of all Riemannian metrics on the body. Moreover, a natural Leibniz rule, which allows to define an objective derivative on contravariant tensor fields given one on covariant tensor fields and vice versa makes the distinction between those of them who are of "Lie type" or of "co-rotational type" useless. We calculate furthermore, for an exhaustive list of objective derivatives found in the literature, the corresponding covariant derivative on Met(B).

On tangent geometry and generalised continuum with defects

“…For linear connections this implies that Γ satisfies the transformation law (64) (properties of the covariant derivative are given in Appendix A.1). This means that from any linear connection N on M = TB one obtains a covariant derivative ∇ on X(B) with Christoffel symbols Γ and vice versa (see [50] for a discussion on covariant derivatives). Remark that all along the paper the terminology Christoffel symbols will refer to the coefficient of a covariant derivative or connection (see (63)) and they are not necessarily related to the Riemannian metric.…”