= Solution
At $(p,q)$, choose neighborhoods on which the two factors admit <local isometries> to open subsets of <Euclidean space>. Their product is a local isometry of the <product Riemannian metric> to the standard Euclidean metric on $\mathbb R^{2n}$. In these product coordinates the metric <matrix> is constant, so every <Christoffel symbol> of the <Levi-Civita connection> is zero. Its <Riemann curvature tensor> therefore vanishes at every point of that neighborhood. Curvature has <tensoriality>, so this is independent of the chosen coordinates; the neighborhoods cover the product.
The splitting provides another useful direct check. Expanding the definition on lifted fields, using zero mixed derivatives and commuting opposite-factor fields, gives the <curvature splitting for a product connection>
$$
R((u_1,u_2),(v_1,v_2))(w_1,w_2)
=\bigl(R^1(u_1,v_1)w_1,R^2(u_2,v_2)w_2\bigr).
$$
<Tensoriality> extends this identity from lifts to arbitrary tangent vectors at a point. Each factor curvature is zero under the stated local-isometry hypothesis. Consequently
$$
\boxed{R^{M_1\times M_2}\equiv0.}
$$
This is a local conclusion about the connection; no global identification of either factor or their product with <Euclidean space> is required.
Back to article page
