Metric compatibility

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

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.