Editing Nonmetricity tensor (section)

==Relation to connection==
We say that a [[connection (mathematics)|connection]] <math>\Gamma</math> is compatible with the metric when its associated covariant derivative of the [[metric tensor]] (call it <math>\nabla^{\Gamma}</math>, for example) is zero, i.e.

:<math> \nabla^{\Gamma}_{\mu}g_{\alpha\beta}=0 .</math>

If the connection is also torsion-free (i.e. totally symmetric) then it is known as the [[Levi-Civita connection]], which is the only one without [[torsion tensor|torsion]] and compatible with the metric tensor. If we see it from a geometrical point of view, a non-vanishing nonmetricity tensor for a metric tensor <math>g</math> implies that the modulus of a vector defined on the [[tangent bundle]] to a certain point <math>p</math> of the manifold, ''changes'' when it is evaluated along the direction (flow) of another arbitrary vector.