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 parallelly transport the metric. Physically, this corresponds to the failure of the metric to preserve angles and lengths under parallel transport.
Definition Let M {\displaystyle M} be a manifold equipped with a metric g {\displaystyle g} , and let ∇ {\displaystyle \nabla } be an affine connection on the tangent bundle T M {\displaystyle TM} . The nonmetricity tensor is defined (some authors use the opposite sign convention) as Q ( X , Y , Z ) := ( ∇ X g ) ( Y , Z ) {\displaystyle Q(X,Y,Z):=(\nabla _{X}g)(Y,Z)} for X , Y , Z {\displaystyle X,Y,Z} arbitrary vector fields. In abstract index notation, this reads Q a b c = ∇ a g b c {\displaystyle Q_{abc}=\nabla _{a}g_{bc}} .
Properties It is manifestly symmetric in its latter two indices due to the symmetry of the metric, and carries n 2 ( n + 1 ) / 2 {\displaystyle n^{2}(n+1)/2} independent components on an n {\displaystyle n} -dimensional manifold. One can additionally define the nonmetricity 1-forms either (and equivalently) by contracting the tensor with a basis 1-form on its first index, or by the exterior covariant derivative D ∇ {\displaystyle D^{\nabla }} associated with the connection ∇ {\displaystyle \nabla } as Q = D ∇ g {\displaystyle \mathbf {Q} =D^{\nabla }g} We say a connection is metric compatible (or sometimes just "metric") if the nonmetricity tensor associated with that connection vanishes. The Levi-Civita connection is the unique metric compatible connection with vanishing torsion.
Use in Physics The triple ( M , g , ∇ ) {\textstyle (M,g,\nabla )} are the data for a metric affine spacetime.
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