= Product affine connection
{title2=$p_1^*\nabla^1\oplus p_2^*\nabla^2$}
Connections $\nabla^1,\nabla^2$ on $TM_1,TM_2$ give a connection on
$$
T(M_1\times M_2)\cong p_1^*TM_1\oplus p_2^*TM_2
$$
by taking the direct sum of their <pullback connections>. In product coordinates its <Christoffel symbols> have the two factor blocks and zero mixed blocks. On arbitrary fields, the ordinary derivatives of their coefficients are still taken in both factors. On fields lifted separately from the factors it satisfies $\nabla_{Y_1+Y_2}(X_1+X_2)=\nabla^1_{Y_1}X_1+\nabla^2_{Y_2}X_2$.
Back to article page
