I know that $\delta\Gamma^\sigma_{\mu\nu}=\frac{1}{2}g^{\sigma\lambda}(\nabla_\nu\delta g_{\mu\lambda}+\nabla_\mu\delta g_{\nu\lambda}-\nabla_\lambda\delta g_{\mu\nu})$.
My question is: if I find a therm like this when calculating Euler-Lagrange equations, can I say that it is a total divergence and using Stokes theorem the variation is zero? (with appropriate boundary conditions)