# Components of Torsion Tensor

So I am trying to calculate the components $T^i\,_{j k}$ relative to a coordinate basis, of the torsion tensor defined as: $$\bf{T(u,v)= \nabla_{u}v-\nabla_v u-[u,v]}$$

Or: $$T= T^i\,_{j k} e_i\otimes\omega^j \otimes \omega^k$$ Now, I have read through a couple resources and found that I can find the components by evaluating my first expression on my basis vectors. I am curious why this is so and if every tensor is like this.

I should add that the $e_\alpha$ and $\omega^\beta$, are my coordinate vectors and the dual vectors (respectively).

If $\{e_i\}$ is a given frame then there are functions such that

$$\nabla_{e_i}e_j=\alpha_{ij}^ke_k,\quad [e_i,e_j]=c_{ij}^ke_k.$$

Then,

$$T(e_i,e_j)=\nabla_{e_i}e_j-\nabla_{e_j}e_i-[e_i,e_j]=\alpha_{ij}^ke_k-\alpha_{ji}^ke_k-c_{ij}^ke_k.$$ That is,

$$T=(\alpha_{ij}^k-\alpha_{ji}^k-c_{ij}^k)e^i\otimes e^j \otimes e_k.$$

Note that since $T$ is a tensor it is enough to compute it on a given basis (of course, this works for any tensor). If you want to evaluate it at $u=u^ie_i,v=v^je_j$ you have

$$T(u,v)=u^iv^jT(e_i,e_j),$$ since, as I have said, it is a tensor.