Torsion tensor

Builds on Affine connection Lie bracket of vector fields Tensor field

Definition

Let $M$ be a smooth manifold with an affine connection $\nabla$ , and let $\mathfrak{X}(M)$ be the space of smooth vector fields on $M$ . The torsion of $\nabla$ is the map $T : \mathfrak{X}(M) \times \mathfrak{X}(M) \to \mathfrak{X}(M)$ given by $T(X, Y) = \nabla_X Y - \nabla_Y X - [X, Y],$ where $[X, Y]$ is the Lie bracket. The connection $\nabla$ is torsion-free if $T = 0$ .

Remarks

$T$ is $C^\infty(M)$ -linear in each argument: for a smooth function $f$ , $T(fX, Y) = f \nabla_X Y - (Yf) X - f \nabla_Y X - f [X, Y] + (Yf) X = f \, T(X, Y)$ . So $T$ is a tensor field of type $(1, 2)$ , and $T(Y, X) = -T(X, Y)$ . In a chart, $T(\partial_i, \partial_j) = \sum_k (\Gamma^k_{ij} - \Gamma^k_{ji}) \partial_k$ , since coordinate vector fields commute. Example: the flat connection of $\R^n$ has $\Gamma^k_{ij} = 0$ , so it is torsion-free. Non-example: on $\R^3$ the connection $\nabla_X Y = D_X Y + X \times Y$ , with $D$ the flat connection and $\times$ the cross product, is a connection but not torsion-free, since $T(X, Y) = 2 \, X \times Y \ne 0$ for independent $X, Y$ .

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Tag	`TW35GWE1XF`
URI	`m2://TW35GWE1XF`
Kind	definition
Writing	W0 stub
MSC	53B05
Defines	Torsion tensor
Uses	Affine connection, Lie bracket of vector fields, Tensor field

Torsion tensor

Definition

Remarks

Reading path

Further reading

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