nonmetricity

Nonmetricity is a property of an affine connection on a differentiable manifold in which the metric tensor is not preserved under parallel transport. For a connection ∇, the nonmetricity tensor is defined by Q_{λ μ ν} = ∇_λ g_{μν}. Since g_{μν} is symmetric, Q_{λ μ ν} is symmetric in μ and ν. If Q_{λ μ ν}=0 for all indices, the connection is metric-compatible; otherwise the connection has nonmetricity.

Nonmetricity describes how the inner product of vectors changes along curves. When Q is nonzero, lengths of vectors and angles between vectors may vary under parallel transport. A connection may also have torsion T; geometries with both nonzero torsion and nonmetricity are studied in metric-affine geometry. A notable special case is Weyl geometry, where ∇_λ g_{μν} = -2 w_λ g_{μν} for a one-form w, leading to scale changes of lengths.

In physics, standard general relativity uses the Levi-Civita connection, which is torsion-free and metric-compatible (Q = 0). The broader framework of metric-affine gravity and related theories allows nonmetricity to play a dynamical role, potentially introducing additional gauge fields or modifying gravitational dynamics. Historically, Weyl introduced nonmetricity in 1918 to unify gravity and electromagnetism, predicting variable lengths; the idea inspired later gauge-theoretic approaches to gravity.