So, I'm having a bit of trouble finding what I've done wrong. If we define the torsion tensor:
$$ T(u, v) = \nabla_u v - \nabla_v u - [u, v] $$
Then I substitute:
$$ T(u, v) = \nabla_u v - \nabla_v u - u(v) + v(u) $$
And then I can write (I'm not bothering to write the $\otimes$ everwhere:
$$ T(u, v) = {\partial v \over \partial x^k} dx^k(u) - {\partial u \over \partial x^k} dx^k(v) - u^ke_k(v) + v^ke_k(u) $$
And then I write:
$$ T(u, v) = {\partial v \over \partial x^k} u^k - {\partial u \over \partial x^k} v^k - u^k{\partial v \over \partial x^k } + v^k{\partial u \over \partial x^k } $$
At this point, it seems that everything just crosses out to give me a torsion tensor of zero. So somewhere along the way, I must have assumed that the te torsion tensor was zero. But I'm not exactly sure where.
I was thinking that perhaps $u(v) = u^k {\partial v \over \partial x^k}$ only makes sense in a torsion free space, but I'm not quite sure why that would be the case. So could anybody point out what I did wrong here.
Thanks in advanced