for my homework I have to prove that inside a Lie derivative of a tensor along a vector field, if the connection is torsion free, I can swap the partial derivatives with covariant ones.
My approach was to show that in general, considering a torsion free connection, the partial derivates and the covariant ones are the same.
During class we defined torsion as $T^\alpha_{\nu\mu}=\frac{\Gamma^\alpha_{[\nu\mu]}}{2}$, with $\Gamma^\alpha_{[\nu\mu]}=\Gamma^\alpha_{\nu\mu}-\Gamma^\alpha_{\mu\nu}$.
I found this definition of torsion, bottom of the fourth page. My idea was to simply say:
Let $\nabla_{[\nu}v_{\mu]}=\partial_{[\nu}v_{\mu]}-\Gamma^\alpha_{[\nu\mu]}v_\alpha \rightarrow \nabla_{[\nu}v_{\mu]}-\partial_{[\nu}v_{\mu]}=-\Gamma^\alpha_{[\nu\mu]}v_\alpha$, given the torsion tensor $T^\alpha_{\nu\mu}=\frac{\Gamma^\alpha_{[\nu\mu]}}{2}$ we get $\nabla_{[\nu}v_{\mu]}-\partial_{[\nu}v_{\mu]}=-2(T^\alpha_{\nu\mu})$ So if the torsion is null: $\nabla_{[\nu}v_{\mu]}=\partial_{[\nu}v_{\mu]}$ And thus I can swap the partial derivatives with covariant ones. Note that the difference regarding the definition encountered in the pdf and in my reasoning (from $\nabla_{[\nu}v_{\mu]}-\partial_{[\nu}v_{\mu]}=-\frac{T^\alpha_{\nu\mu}}{2}$ to $\nabla_{[\nu}v_{\mu]}-\partial_{[\nu}v_{\mu]}=-2(T^\alpha_{\nu\mu})$) is due to the different definiton of torsion (found at half of page four in the document), and since my professor used the one I gave at the start I'll be using that.
I however have two questions, first, is my reasoning correct? If not, what's the right approach? Would you consider this an acceptable answer?