# Changes

## Torsion of a linear connection

, 17:48, 6 January 2012
Tensoriality
The torsion map is a $(1,2)$ tensor. It is tensorial in both $X$ and $Y$. This means that the value of the torsion of a connection for two vector fields at a point depends only on the values of the vector fields at that point. In other words, $\tau(\nabla)(X,Y)$ at $p$ depends on $\nabla, X(p), Y(p)$ only and does not depend on how $X$ and $Y$ behave elsewhere on the manifold.

===Antisymmetry===

{{further|[[Torsion is antisymmetric]]}}

We have that the torsion tensor is antisymmetric, i.e., we have:

$\tau(\nabla)(Y,X) = -\tau(\nabla)(X,Y)$

Equivalently, we have that:

$\tau(\nabla)(X,X) = 0$