Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 115 / 4 / a / Solution

The solder form on $X$ is the $TX$-valued one-formIn a coordinate frame it is $\theta =d{x}^i\otimes {\partial }_i$. The torsion form of $\nabla$ isEquivalently, for vector fields $U,V$,Writingand using $[{\partial }_i,{\partial }_j]=0$ givesHence $\nabla$ is torsion-free exactly whenfor every $i,j,k$.