Nonmetricity tensor
In mathematics, the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor. It can be interpreted as the failure of a connection to parallely transport the metric. Physically, this corresponds to the failure of the metric to preseve angles and lengths under parallel transport.
Definition
Let be a manifold equipped with a metric, and let be an affine connection on the tangent bundle. The nonmetricity tensor is defined asfor arbitrary vector fields. In abstract index notation, this reads.Properties
It is manifestly symmetric in its latter two indices due to the symmetry of the metric, and carries independent components on an -dimensional manifold.One can additionally define the nonmetricity 1-forms either by contracting the tensor with a basis 1-form on its first index, or by the exterior covariant derivative associated with the connection asWe say a connection is metric compatible if the nonmetricity tensor associated with that connection vanishes.
The Levi-Civita conneciton is the unique metric compatible connection with vanishing torsion.