Metric compatibility

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.

References

Rodrigues, W. A.; Fernández, V. V.; Moya, A. M. (2005). "Metric compatible covariant derivatives". arXiv:math/0501561.
Wald, Robert M. (1984), General Relativity, University of Chicago Press, ISBN 0-226-87033-2