metricaffine

Metric-affine geometry, sometimes called metricaffine geometry, is the study of differentiable manifolds equipped with two independently specified structures: a metric tensor g and an affine connection ∇. Unlike in Riemannian geometry, where the connection is required to be the Levi-Civita connection of g (torsion-free and metric-compatible), metric-affine geometry allows ∇ to be independent of g. This generality permits nonzero torsion and nonmetricity, broadening the range of geometric phenomena that can be modeled.

The key objects in metric-affine geometry are torsion and nonmetricity. Torsion T is defined by T(X,Y) = ∇_X Y − ∇_Y X − [X,Y], measuring the failure of the connection to be symmetric. Nonmetricity Q captures how the metric changes under parallel transport: Q(X,Y,Z) = (∇_X g)(Y,Z). In components, the connection coefficients Γ^λ_{ μν } and the metric g_{μν} are treated as independent, subject to the usual compatibility with the underlying smooth structure.

Special cases illustrate the framework. If Q = 0 and T = 0, one recovers Riemannian geometry with the Levi-Civita connection. If ∇ is metric-compatible (Q = 0) but possesses torsion, the setting relates to Einstein–Cartan theory. If torsion vanishes but nonmetricity is present, one arrives at Weyl-type or more general nonmetric geometries. In physics, metric-affine gravity treats g and Γ as independent dynamical fields, often studied via Palatini variations, leading to field equations that couple curvature, torsion, and nonmetricity to matter.

In mathematics and theoretical physics, metricaffine geometry provides a unified language for affine connections with arbitrary torsion and nonmetricity, encompassing a broad class of geometric and gravitational theories.