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.
For a metric-compatible affine connection, its difference from the Levi-Civita connection is the contorsion tensor. It is determined by the torsion tensor. In derivative-last notation let , the negative of geometric torsion, and lower the last slot with the metric tensor. Then . This is the actual correction added to the Levi-Civita connection; a convention writing has .
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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.