= Curvature splitting for a product connection
For a <product affine connection>, its curvature satisfies
$$
R((u_1,u_2),(v_1,v_2))(w_1,w_2)
=(R^1(u_1,v_1)w_1,R^2(u_2,v_2)w_2).
$$
Expand the defining derivative commutator on lifted <vector fields>. Opposite-factor fields commute and have zero mixed covariant derivatives, leaving exactly the two factor curvatures. Tensoriality gives the formula for arbitrary tangent vectors. Thus the product connection is flat if the two factors are flat; for a <product Riemannian metric> this applies to its <Levi-Civita connection>.
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