Metric compatibility

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 $g_{ab}$, a covariant derivative is said to be compatible with the metric if the following condition is satisfied:

$\nabla _{c}\,g_{ab}=0.$

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, $\nabla$ and $\nabla '$, there exists a tensor for transforming from one to the other:

$\nabla _{a}x_{b}=\nabla _{a}'x_{b}-{C_{ab}}^{c}x_{c}.$

If the space is also torsion-free, then the tensor ${C_{ab}}^{c}$ is symmetric in its first two indices.

References

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