Suppose $M_i$ are smooth manifolds and denote by $C^\infty(M_i)$, $\mathfrak{X}(M_i)$ the space of smooth functions and smooth vector fields on $M_i$. How far from true is the statement
\begin{equation}
\mathfrak{X}(M_1 \times M_2) = C^\infty(M_1\times M_2) (\mathfrak{X}(M_1)\oplus \mathfrak{X}(M_2))?
\end{equation}
Also, can $C^\infty(M_1\times M_2)$ be expressed in terms of $C^\infty(M_1)$, $C^\infty(M_2)$?
Best Answer
One has $T(M_1\times M_2) \simeq \pi_1^*(TM_1) \oplus \pi_2^*(TM_2)$ where $\pi_i\colon M_1\times M_2 \to M_i$ is the natural projection map. It follows that $\mathfrak{X}(M_1\times M_2)$ is generated, as a $\mathcal{C}^{\infty}(M_1\times M_2)$-module, by elements of the form $\pi_1^*(X_1)\oplus \pi_2^*(X_2)$ with $X_i \in \mathfrak{X}(M_i)$, which we would rather write $X_1\oplus X_2$, $(X_1,X_2)$ or even $X_1+X_2$ if the context is clear.
However, one cannot express $\mathcal{C}^{\infty}(M_1\times M_2)$ in such a way that it only involves $\mathcal{C}^{\infty}(M_i)$ separately, at least not in a nice way.