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 <math>M</math> be a manifold equipped with a metric <math>g</math>, and let <math>\nabla</math> be an affine connection on the tangent bundle <math>TM</math>. The nonmetricity tensor is defined (some authors use the opposite sign convention) as<math display="block">Q(X,Y,Z) := (\nabla_X g)(Y,Z)</math>for <math>X,Y,Z</math> arbitrary vector fields. In abstract index notation, this reads <math>Q_{abc} = \nabla_a g_{bc}</math>.
Properties
It is manifestly symmetric in its latter two indices due to the symmetry of the metric, and carries <math>n^2 (n+1)/2</math> independent components on an <math>n</math>-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 <math>D^\nabla</math> associated with the connection <math>\nabla</math> as<math display="block">\mathbf{Q} = D^\nabla g</math>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 <math display="inline">(M,g,\nabla)</math> are the data for a metric affine spacetime