RiemannCartan

Riemann-Cartan geometry is a generalization of Riemannian geometry in which a differentiable manifold is equipped with a metric g and an affine connection that is metric-compatible but may have torsion. In this setting, parallel transport is influenced by both curvature and torsion, and the geometry is often described as a Riemann–Cartan or RC structure. When torsion vanishes, the framework reduces to ordinary Riemannian geometry used in general relativity.

Mathematically, the affine connection Γ^λ_{ μν } need not be symmetric in its lower indices, and the torsion tensor is defined by T^λ_{ μν } = Γ^λ_{ μν } − Γ^λ_{ νμ }. The condition ∇_λ g_{μν} = 0 expresses metric-compatibility. The connection can be decomposed as Γ^λ_{ μν } = {λ}_{ μν } + K^λ_{ μν }, where {λ}_{ μν } are the Christoffel symbols of the metric, and K^λ_{ μν } is the contortion tensor related to torsion. Curvature is built from Γ in the usual way, giving a curvature tensor with modified Bianchi identities due to torsion.

In physics, RC geometry provides the geometric setting for Einstein–Cartan theory, where torsion is sourced by spin density of matter. Torsion couples to intrinsic spin and can avoid certain singularities in high-density regimes, though it is typically non-propagating and vanishes in vacuum for standard formulations. RC geometry also underpins broader gauge-theoretic approaches to gravity, such as Poincaré gauge theory, and is closely related to formulations using tetrads or vielbeins.