Nonmetricity tensor

In mathematics, the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor.[1][2] It is therefore a tensor field of order three. It vanishes for the case of Riemannian geometry and can be used to study non-Riemannian spacetimes.[3]


By components, it is defined as follows.[1]

It measures the rate of change of the components of the metric tensor along the flow of a given vector field, since

where is the coordinate basis of vector fields of the tangent bundle, in the case of having a 4-dimensional manifold.

Relation to connection

We say that a connection is compatible with the metric when its associated covariant derivative of the metric tensor (call it , for example) is zero, i.e.

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 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 implies that the modulus of a vector defined on the tangent bundle to a certain point of the manifold, changes when it is evaluated along the direction (flow) of another arbitrary vector.


  1. Hehl, Friedrich W.; McCrea, J. Dermott; Mielke, Eckehard W.; Ne'eman, Yuval (July 1995). "Metric-affine gauge theory of gravity: field equations, Noether identities, world spinors, and breaking of dilation invariance". Physics Reports. ScienceDirect. 258: 1. arXiv:gr-qc/9402012. doi:10.1016/0370-1573(94)00111-F. Retrieved 2019-04-29.
  2. Kopeikin, Sergei; Efroimsky, Michael; Kaplan, George (2011), Relativistic Celestial Mechanics of the Solar System, John Wiley & Sons, p. 242, ISBN 9783527408566.
  3. Puntigam, Roland A.; Lämmerzahl, Claus; Hehl, Friedrich W. (May 1997). "Maxwell's theory on a post-Riemannian spacetime and the equivalence principle". Classical and Quantum Gravity. IOP Publishing. 14: 1347. arXiv:gr-qc/9607023. doi:10.1088/0264-9381/14/5/033. Retrieved 2019-04-29.

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