In differential geometry, the torsion tensor is a tensor that is associated to any affine connection. The torsion tensor is a bilinear map of two input vectors ${\displaystyle X,Y}$, that produces an output vector ${\displaystyle T(X,Y)}$ representing the displacement within a tangent space when the tangent space is developed (or "rolled") along an infinitesimal parallelogram whose sides are ${\displaystyle X,Y}$. It is skew symmetric in its inputs, because developing over the parallelogram in the opposite sense produces the opposite displacement, similarly to how a screw moves in opposite ways when it is twisted in two directions.

Torsion is particularly useful in the study of the geometry of geodesics. Given a system of parametrized geodesics, one can specify a class of affine connections having those geodesics, but differing by their torsions. There is a unique connection which absorbs the torsion, generalizing the Levi-Civita connection to other, possibly non-metric situations (such as Finsler geometry). The difference between a connection with torsion, and a corresponding connection without torsion is a tensor, called the contorsion tensor. Absorption of torsion also plays a fundamental role in the study of G-structures and Cartan's equivalence method. Torsion is also useful in the study of unparametrized families of geodesics, via the associated projective connection. In relativity theory, such ideas have been implemented in the form of Einstein–Cartan theory.

Let M be a manifold with an affine connection on the tangent bundle (aka covariant derivative) ∇. The torsion tensor (sometimes called the Cartan (torsion) tensor) of ∇ is the vector-valued 2-form defined on vector fields X and Y by

where [X, Y] is the Lie bracket of two vector fields. By the Leibniz rule, T(fX, Y) = T(X, fY) = fT(X, Y) for any smooth function f. So T is tensorial, despite being defined in terms of the connection which is a first order differential operator: it gives a 2-form on tangent vectors, while the covariant derivative is only defined for vector fields.

The components of the torsion tensor ${\displaystyle T^{c}{}_{ab}}$ in terms of a local basis (e₁, ..., e_n) of sections of the tangent bundle can be derived by setting X = e_i, Y = e_j and by introducing the commutator coefficients γ^k_ije_k := [e_i, e_j]. The components of the torsion are then

Here ${\displaystyle {\Gamma ^{k}}_{ij}}$ are the connection coefficients defining the connection. If the basis is holonomic then the Lie brackets vanish, ${\displaystyle \gamma ^{k}{}_{ij}=0}$. So ${\displaystyle T^{k}{}_{ij}=2\Gamma ^{k}{}_{[ij]}}$. In particular (see below), while the geodesic equations determine the symmetric part of the connection, the torsion tensor determines the antisymmetric part.

The torsion form, an alternative characterization of torsion, applies to the frame bundle FM of the manifold M. This principal bundle is equipped with a connection form ω, a gl(n)-valued one-form which maps vertical vectors to the generators of the right action in gl(n) and equivariantly intertwines the right action of GL(n) on the tangent bundle of FM with the adjoint representation on gl(n). The frame bundle also carries a canonical one-form θ, with values in Rⁿ, defined at a frame u ∈ F_xM (regarded as a linear function u : Rⁿ → T_xM) by

where π  : FM → M is the projection mapping for the principal bundle and π∗ is its push-forward. The torsion form is then