Inmathematics, thenonmetricity tensor indifferential geometry is thecovariant derivative of themetric tensor. It can be interpreted as the failure of a connection toparallely transport the metric. Physically, this corresponds to the failure of the metric to preseve angles and lengths under parallel transport.

Let${\displaystyle M}$ be amanifold equipped with ametric${\displaystyle g}$, and let${\displaystyle \mathrm{\nabla }}$ be anaffine connection on thetangent bundle${\displaystyle TM}$. The nonmetricity tensor is defined (some authors use the opposite sign convention) as$${\displaystyle Q(X,Y,Z):=({\mathrm{\nabla }}_{X}g)(Y,Z)}$$for${\displaystyle X,Y,Z}$ arbitraryvector fields. Inabstract index notation, this reads${\displaystyle Q_{abc}={\mathrm{\nabla }}_a{g}_{bc}}$.

It is manifestly symmetric in its latter two indices due to the symmetry of the metric, and carries${\displaystyle n^2(n+1)/2}$ independent components on an${\displaystyle n}$-dimensional manifold.

One can additionally define thenonmetricity 1-forms either (and equivalently) by contracting the tensor with a basis 1-form on its first index, or by theexterior covariant derivative${\displaystyle D^{\mathrm{\nabla }}}$ associated with the connection${\displaystyle \mathrm{\nabla }}$ as^[1]$${\displaystyle \mathbf{Q}=D^{\mathrm{\nabla }}g}$$We say a connection is metric compatible (or sometimes just "metric") if the nonmetricity tensor associated with that connection vanishes.

TheLevi-Civita conneciton is the unique metric compatible connection with vanishingtorsion.

The triple$(M,g,\mathrm{\nabla })$ are the data for a metric affine spacetime^[1].

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