Covariant Derivatives and Curvature

Partial derivatives of tensor fields are generally not tensor fields. The concept of a covariant derivative is a modification of the concept of a partial derivative, defined so that covariant derivatives of tensor fields are still tensor fields. Curvature tensors are defined in terms of covariant derivatives. General relativity is formulated with the help of a special covariant derivative that is metric-compatible and torsion-free.

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Revision history

2023-12-11 (added a paragraph about the symmetry of the Ricci tensor, added another terminology variation)
2023-12-09 (split the section about the curvature tensor into smaller sections, revised it to use a fixed sign convention for the Ricci tensor that seems to be standard in the physics literature, moved the curvature-of-sphere sign-check to a separate article devoted to sign conventions)
2022-06-11 (updated references to other articles in this series to link to html abstract pages instead of to pdfs. Didn't change the version number/date)
2022-02-06 (fixed incorrect statement about metric-dependence in the context of sign conventions)
2022-02-05 (first version)

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updated 2025-02-25

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Covariant Derivatives and Curvature

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