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.
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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.