Difference between $\mathfrak{X}(M)\otimes_{\mathbb{R}}\mathfrak{X}(M)$ and $\Gamma(TM\boxtimes TM)$ Let $M$ be a smooth manifold and $\mathfrak{X}(M)$ the real vector space consisting of vector fields over $M$. Let $TM\boxtimes TM$ be the vector bundle over $M\times M$ whose fiber at $(x,y)\in M\times M$ is $T_xM\otimes T_yM$. Finally, let $\Gamma(TM\boxtimes TM)$ denote the sections of this vector bundle.
Is there any essential difference between the tensor product of real vector spaces $\mathfrak{X}(M)\otimes_{\mathbb{R}}\mathfrak{X}(M)$ and the real vector space $\Gamma(TM\boxtimes TM)$?
There seems to be a well-defined $\mathbb{R}$-linear map $\phi:\mathfrak{X}(M)\otimes_{\mathbb{R}}\mathfrak{X}(M)\rightarrow \Gamma(TM\boxtimes TM)$ given by $\phi(\sum_k X_k\otimes Y_k)(x,y)=\sum_k X_k(x)\otimes Y_k(y)$. Provided that the bundle $TM\boxtimes TM$ is paracompact and can be covered by finitely many bundle charts, I think that you can use a partition of unity to show that $\phi$ is surjective. I'm having a bit of trouble proving it's injective though. I also can't think of a non-trivial element of $\mathfrak{X}(M)\otimes_{\mathbb{R}}\mathfrak{X}(M)$ that would map to the zero section. 
 A: Let $M=\def\RR{\mathbb R}\RR$. Let $\def\X{\mathfrak X}\X(M)$ be the vector space of tangent fields to $M$, and let $\partial\in\X(M)$ be any vector field which is never-zero, so that $\{\partial\}$ is a $C^\infty(M)$-basis of the $C^\infty(M)$-module $\X(M)$, and the map $$\alpha:f\in C^\infty(M)\mapsto f\partial\in\X(M)$$ is an isomorphism.
The bundle $TM$ is a trivial line bundle on $M$. It follows that $TM\boxtimes TM$ is also a trivial line bundle on $M\times M$. We can view $\partial$ as a section $M\to TM$ which generates at each point $TM$, and then $(p,q)\in M\times M\mapsto\partial_p\otimes\partial_q\in (TM\boxtimes TM)_{(p,q)}$ is a section of $TM\boxtimes TM$ which generates at each point, let us call it $D$. It follows that $$\beta:f\in C^{\infty}(M\times M)\mapsto fD\in\Gamma(TM\boxtimes TM)$$ is a vector space isomorphism.
Now your map $\X(M)\otimes_\RR\X(M)\to\Gamma(TM\otimes TM)$ can be conjugated by the maps $\alpha$ and $\beta$ to give a map $$C^\infty(M)\otimes_\RR C^\infty(M)\to C^\infty(M\times M).$$
Make it explicit and see that it is not an isomorphism.

The problem of deciding when a function is in the image of the above map is a very classical one. Here is a necessary condition: suppose $h(x,y)=\sum_{\ell=1}^nf_i(x)g_i(y)$ can be written as a sum of $n$ products of decomposable functions. Then the determinant $$\mathcal W_n(h)=\det\left(\frac{\partial^{i+j}h}{\partial x^i\partial y^j}\right)_{i,j=0,\dots,n}$$ vanishes. This determinant is called a Wronksian, if I recall correctly (although it is not the same as the Wronskian one finds in the context of linear ODEs —it is related, of course)
Now, a little work will show that $\mathcal W_n(\exp(xy))\neq0$ for all $n\geq1$, so that $\exp(xy)$ is not in the image of our map. (In fact, $\mathcal W_n(\exp(xy))$ is $\exp((n+1)xy)$ times the Vandermonde determinant for $1$, $2$, $\dots$, $n+1$.)
