Abstract
A coordinate-invariant description of a Riemannian manifold is known to be furnished by the curvature tensor and a finite number of its covariant derivatives relative to a field of orthogonal frames. These tensors are closer to measurements than the metrical tensor is. The present article discusses this description's usefulness in general relativity and the redundancy among the curvature tensor and its derivatives.
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Karlhede, A. On a coordinate-invariant description of Riemannian manifolds. Gen Relat Gravit 12, 963–970 (1980). https://doi.org/10.1007/BF00757367
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DOI: https://doi.org/10.1007/BF00757367