Metric compatibility

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This article is about the concept in Riemannian geometry. For the typographic concept, see Typeface#Font metrics.

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In mathematics, given a metric tensor , a covariant derivative is said to be compatible with the metric if the following condition is satisfied:

Although other covariant derivatives may be supported within the metric, usually one only ever considers the metric-compatible one. This is because given two covariant derivatives, and , there exists a tensor for transforming from one to the other:

If the space is also torsion-free, then the tensor is symmetric in its first two indices.


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