Question: How to simplify the following local expression: \begin{equation}\tag{*} g^{ij} \Big( \partial_j (\partial_i V^k + \Gamma_{im}^k V^m) - \Gamma_{ij}^l (\partial_l V^k + \Gamma_{lm}^k V^m) \Big), \end{equation} where $V$ is a vector field on a Riemannian manifold $(M,g)$, $\Gamma$ is the Christoffel symbols.
Motivation: I am trying to get through the old paper of Dohrn and Guerra, which has the following quantity in its eqaution (12): \begin{equation}\tag{**} g^{ij} \nabla_{\partial_j} \nabla_{\partial_i} V, \end{equation} where $\nabla$ is the Riemannian covariant derivative. According to my derivation from the preceding text of the paper, the quantity $(**)$ should coorespond to the local expression $(*)$. However, a simple application of definitions gives the local expression of $(**)$ as follows, \begin{equation} g^{ij} \nabla_{\partial_j} \nabla_{\partial_i} V = g^{ij} \Big( \partial_j (\partial_i V^k + \Gamma_{im}^k V^m) + \Gamma_{jm}^k (\partial_i V^m + \Gamma_{il}^m V^l) \Big) \partial_k, \end{equation} which does not coincide with (*). So I strongly suspect that the expression (**) in the paper is not correct.
But I still want to know if it is possible to simplify the local expression (*) to a quantity with a global expression, which may be similar to (**) ? TIA...
Expanding the outer covariant operator gives us: $$ g^{ij} \nabla_{\partial_j} (\nabla_{\partial_i} V^k) = \\g^{ij} \Big( \partial_j (\nabla_{\partial_i} V^k) + \Gamma_{jm}^k (\nabla_{\partial_i} V^m) - \Gamma_{jk}^i (\nabla_{\partial_i} V^k) \Big) $$ This has a positive signed Christoffel symbol for the inner contravariant index $k$ and a negative signed one for the covariant inner index $i$. This leads to the full expansion: $$ g^{ij} \nabla_{\partial_j} (\nabla_{\partial_i} V^k) = \\g^{ij} \Big( \partial_j (\partial_i V^k + \Gamma_{im}^k V^m) + \Gamma_{jm}^k (\partial_i V^m + \Gamma_{il}^m V^l) - \Gamma_{jk}^i (\partial_i V^k + \Gamma_{im}^k V^m) \Big) $$ which looks like a combination of the two expressions in your post.