 Solution

The product Riemannian metric has block matrixIn the given Christoffel symbols formula, every mixed coefficient vanishes. Off-diagonal metric entries are zero, ${\partial }_{y^{\alpha }}{g}_{ij}=0$, and ${\partial }_{x^i}{g}_{\alpha \beta }=0$. These facts give ${\Gamma }_{i\alpha }^A=0$ for every output index $A$, as well as ${\Gamma }_{ij}^{\alpha }=0$ and ${\Gamma }_{\alpha \beta }^i=0$. The remaining pure blocks are precisely the Christoffel symbols of the respective factor metrics.

For the lifted fields of (i), ${\partial }_{x^i}{Y}^{\alpha }=0$. ConsequentlyThus $\boxed{{\nabla }_{X}Y=0}$. The same calculation gives ${\nabla }_{Y}X=0$; differentiating two fields from one factor gives the lifted factor covariant derivative. This identifies the product Levi-Civita connection with the product affine connection and proves the curvature splitting used in the root solution.