One can define the connection laplacian on $TM$ as $$ \Delta X := \sum_i \nabla_i \nabla_i X - \nabla_{\nabla_i \partial_i} X $$
where $\{\partial_i\}$ give an orthonormal basis at a point, induce by coordinates. If we further assume these are parallel (e.g. geodesic normal coordinates), then $$ \Delta X := \sum_i \nabla_i \nabla_i X = \sum_i X^k_{ii} \partial_k + X^k \nabla_i \nabla_i \partial_k $$ Is there a name or reduction for the connection laplacian on basis elements, e.g. $V_k := \sum_i \nabla_i \nabla_i \partial_k$? In terms of christoffels, we can write $$ V_k = \sum_i \nabla_i \nabla_i \partial_k = (\sum_i \Gamma_{ik,i}^j) \partial_j $$ So equivalently, can this sum of derivatives of christoffels be reduced in general?