---
aliases: ["torsion"]

---

The torsion measures how closely the Lie bracket of vector fields can be recovered from the [Riemannian Geometry](Unsorted/Riemannian%20Geometry.md).

Definition: for $\nabla$ a [Riemannian Geometry](Unsorted/Riemannian%20Geometry.md),

$$
\tau^{\nabla}(X, Y) := \nabla_{X} Y-\nabla_{Y} X-[X, Y]
$$