Definition:Torsion Tensor

Definition

Let $M$ be a smooth manifold.

Let $TM$ be the tangent bundle of $M$.

Let $\nabla$ be a connection in $TM$.

Let $\sqbrk {\cdot, \cdot}$ be the Lie bracket.

Let $\map {\mathfrak{X}} M$ be the space of smooth vector fields on $M$.

Let $\tau : \map {\mathfrak{X}} M \times \map {\mathfrak{X}} M \to \map {\mathfrak{X}} M$ be a tensor field such that:

$\forall X, Y \in \map {\mathfrak{X}} M : \map \tau {X, Y} := \nabla_X Y - \nabla_Y X - \sqbrk {X, Y}$

Then $\tau$ is called the torsion tensor.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 5$: The Levi-Civita Connection. Symmetric Connections