The torsion tensor is a mathematical object that arises in differential geometry and is used in the context of manifold theory, especially in connection with affine connections and Riemannian geometry. It provides a way to describe the twisting or non-symmetries of a connection on a manifold. ### Definition In general, a connection on a manifold defines how to compare tangent vectors at different points, allowing us to define notions such as parallel transport and differentiation of vector fields.
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The torsion of an affine connection is the displayed antisymmetric tensor. It measures the failure of the antisymmetrized covariant derivative to agree with the Lie bracket of vector fields. With derivative-last coefficients , its components are . Some conventions use the negative component tensor. Geodesic equations depend on the symmetric connection coefficients, so they alone cannot detect torsion.