= Product Riemannian metric
{title2=$g_1\oplus g_2$}
The product of <Riemannian manifolds> has metric
$$
g=p_1^*g_1+p_2^*g_2,\qquad g((u_1,u_2),(v_1,v_2))=g_1(u_1,v_1)+g_2(u_2,v_2).
$$
The tangent summands are orthogonal, and positivity follows because a nonzero product tangent vector has a nonzero component. Its <Levi-Civita connection> is the <product affine connection> of the two factor <Levi-Civita connections>: mixed torsion is zero and metric compatibility follows from independence of the two coordinate sets.
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