# Metric compatibility

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.