I am asking this in the context of differential geometry (specifically Riemannian).
When the Levi-Civita Connection is defined, we require that the torsion tensor is 0, which in local coordinates translates to the requirement that $\Gamma_{ij}^{k} = \Gamma_{ji}^{k}$; which is the covariant derivative version of saying partial derivatives commute: $\Delta_{\partial_i}(\partial_j)=\Delta_{\partial_j}(\partial_i)$.\nabla_{\partial_i}(\partial_j)=\nabla_{\partial_j}(\partial_i)$.
This is obviously true in the Euclidian settings, and I understand all the details of the proofs. But why is this such an essential property? Why does this capture our intuitive sense of derivatives?
