The torsion measures how closely the Lie bracket of vector fields can be recovered from the connection.
Definition: for \(\nabla\) a connection,
\begin{align*} \tau^{\nabla}(X, Y) := \nabla_{X} Y-\nabla_{Y} X-[X, Y] \end{align*}
The torsion measures how closely the Lie bracket of vector fields can be recovered from the connection.
Definition: for \(\nabla\) a connection,
\begin{align*} \tau^{\nabla}(X, Y) := \nabla_{X} Y-\nabla_{Y} X-[X, Y] \end{align*}