Given a metric tensor and an affine connection, measures failure of metric compatibility. It is symmetric in . Some authors instead define nonmetricity as , so formulas must specify the sign. Together with the torsion tensor, it determines the difference from the Levi-Civita connection.
With derivative-last connection coefficients and , the displayed tensor is the symmetric-in- correction associated with the nonmetricity tensor. Raising its final slot gives the correction to connection coefficients. It vanishes for metric compatibility. Reversing the sign used to define nonmetricity reverses this formula.
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The nonmetricity tensor is a mathematical object used in the context of a generalization of the theory of gravity, particularly in modifications of general relativity, such as in theories of metric-affine geometry. In differential geometry, the notion of nonmetricity is concerned with the way lengths and angles change under parallel transport. In the context of a connection on a manifold, the nonmetricity tensor is defined as the tensor that measures the failure of the connection to preserve the metric tensor during parallel transport.