Torsion Tensor

The torsion tensor is a tensor associated with an affine connection on a manifold that measures the failure of infinitesimal parallelograms formed by parallel transport to close up, equivalently the antisymmetric part of the connection's coefficients. A connection with zero torsion, such as the Levi-Civita connection used in standard Riemannian geometry, is called torsion-free, and most classical differential geometry works with torsion-free connections. Nonzero torsion appears in more general geometric frameworks, including certain formulations of gravitational theory that generalize general relativity to allow spacetime torsion alongside curvature.