Objective rates as covariant derivatives on the manifold of Riemannian metrics
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2021
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| _version_ | 1866911959506485248 |
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| author | Kolev, Boris Desmorat, Rodrigue |
| author_facet | Kolev, Boris Desmorat, Rodrigue |
| contents | 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). |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2106_01126 |
| institution | arXiv |
| publishDate | 2021 |
| record_format | arxiv |
| spellingShingle | Objective rates as covariant derivatives on the manifold of Riemannian metrics Kolev, Boris Desmorat, Rodrigue Differential Geometry Medical Physics 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). |
| title | Objective rates as covariant derivatives on the manifold of Riemannian metrics |
| topic | Differential Geometry Medical Physics |
| url | https://arxiv.org/abs/2106.01126 |