So, I know that tangent bundle of a product manifold $M \times N$ splits in a sum
$$
T_{(x,y)}(M \times N) = T_xM \oplus T_yN,
$$
so that is obvious that the sum $X \oplus Y$ of smooth vector fields $X \in \mathcal{T}(M)$ and $Y \in \mathcal{T}(N)$ is a smooth vector field of $M \times N$. I've been told that, though not every vector field in $\mathcal{T}(M \times N)$ is a sum, locally one can always find one such decomposition, which will in turn be unique due the fact the sum is a direct one.
How can I show that this decomposition exists locally? More than that, if $X = X_1 + X_2$ is the decomposition, is there a way the express the coordinate functions of $X_1$ and $X_2$ in terms of those of $X$?
First I thought about taking two frames that locally spam $TM$ and $TN$ and write down $X$ using them, but then the coordinate functions are of the form $X^i: M \times N \to \mathbb R$, and the vector field components in each subspace are not exactly fields of $M$ and $N$ because their coordinate functions don't have the right domains. Is there another better way to see this decomposition holds locally?
Best Answer
This is totally false. Indeed, a vector field which is locally of the form $X\oplus Y$ is also globally of that form (the local $X$'s and $Y$'s will always glue together, since they are unique if they exist). Not every vector field on $M\times N$ has this form, since the $TM$ component of a vector field may change between points with the same $M$ coordinate.
For a really simple explicit example, let $M=N=\mathbb{R}$ and identify vector fields on $M$ and $N$ with functions $\mathbb{R}\to\mathbb{R}$ and vector fields on $M\times N$ with functions $\mathbb{R}^2\to\mathbb{R}^2$. Then given two such functions $X,Y:\mathbb{R}\to\mathbb{R}$, their sum $X\oplus Y$ is identified with the function $F(s,t)=(X(s),Y(t))$. Obviously not every smooth function $\mathbb{R}^2\to\mathbb{R}^2$ has this form (e.g., the function $F(s,t)=(t,s)$ does not).
Note that the post you link to does not claim any such thing. Instead it claims a vector field can locally be written as a linear combination of vector fields of the form $X\oplus 0$ or $Y\oplus 0$ with coefficients that are smooth functions on $M\times N$. Those coefficients are crucial, since they can be smooth functions which genuinely live on the product and don't come from either coordinate alone. Allowing such coefficients, the conclusion is trivial. Indeed, choosing local coordinates on $M\times N$ that are a product of local coordinates on $M$ and local coordinates on $N$, every vector field on $M\times N$ is locally a linear combination of the coordinate vector fields (with smooth functions as coefficients). The coordinate vector fields each have the form $X\oplus 0$ or $0\oplus Y$ (the coordinate vector fields for coordinates which come from $M$ are just $X\oplus 0$ where $X$ is the corresponding coordinate vector field on $M$, and similarly for the coordinates that come from $N$).