curly fields

Potentials \(\Phi\) exist for conservative fields \(\vec{g}\), provided that the field has a vanishing rotation, \(\nabla\times\vec{g}=0\), and then \(\vec{g}\) can be computed by \(\vec{g}=-\nabla\Phi\). Can you draw an analogy to the metric compatibility condition \(\nabla_\mu g_{\alpha\beta}=0\) imposing a condition on a connection \(\Gamma^\mu_{\alpha\beta}\), or to the condition for the absence of torsion, \(\Gamma^\mu_{\alpha\beta} = \Gamma^\mu_{\beta\alpha}\)?