Regarding the isomorphism $T_{(x,y)}(X \times Y)= T_xX\times T_yY$ between the tangent spaces of two manifolds $X$ and $Y$, is it true that $$\mathfrak{X}(X \times Y)=\mathfrak{X}(X) \oplus \mathfrak{X}(Y),$$ where $\mathfrak{X}(-)$ denotes the module of vector fields?
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See if you can use the map used to prove the isomorphism to show something about the maps on the module level. – Osama Ghani Oct 27 '17 at 11:51
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@A.Goodier The theorem that you have refered to have different meaning than $\mathfrak{X}(M \times N) = \mathfrak{X}(M) \oplus \mathfrak{X}(N)$. – Kelvin Lois Jun 04 '20 at 04:32