Let $X, Y$ be vector fields on a manifold $M$. Show, that $XY$ is not a vector field

Let $X, Y$ be vector fields on a manifold $M$. Show, that $XY$ is not a vector field

My attempt

My idea would be to directly show, that

$$XY (fg) \neq (XY f)g + f(XY g) \quad \lor \quad XY (\alpha f + \beta g) \neq \alpha XY f + \beta XY g$$

I am not sure at writing out $XY (fg)$

$$XY (fg) = X ( Y (fg)) = X ((Yf)g + f(Yg)) = X((Yf)g) + X(f(Yg)) = $$

$$ = (X (Y f))g + (Yf)Xg + (Xf)(Yg) + f (X(Yg)) = (XYf)g + (Yf)Xg + (Xf)Yg + f(XYg)$$

So we get an additional $(Yf)Xg + (Xf)Yg$, thus $XY (fg) \neq (XY f)g + f(XY g)$

Which is enough to show that $XY$ is not a vector field. Are my calculations alright? I'm not sure especially because at the end, I simply ignore the brackets i.e. $(X(Yf))g$ becomes $(XYf)g$ so it can fit into our first assumption. I don't know if we can do that

Solution 1:

Just to make my first hint explicit: The claim you are trying to prove is (in general) false. As an example, take any smooth manifold $M$ and the vector fields $X=0$ and $Y$ arbitrary. Then for every smooth function $f$ on $M$, $$ XY(f)=0, $$ which implies that in this example $XY=X$, is a vector field.

In case you simply forgot to assume that both vector fields $X, Y$ are nonzero, here is another example. Suppose that $M$ has two connected components $M_1, M_2$ and you have vector fields $X, Y$ on $M$ such that $X$ restricts to $0$ on $M_1$ and $Y$ restricts to $0$ on $M_2$. Then again $$ XY(f)=0 $$ for every smooth function $f$ on $M$. Hence, again, $XY$ is the zero vector field.

So, what are the missing assumptions in your question?

Edit. Here a correct statement (which, IMHO, your professor should have assigned):

For every manifold nonempty $M$ and two vector field $X, Y$ on $M$ the following are equivalent:

1. The composition $XY$ is a vector field.

2. The composition $XY$ is zero vector field.

3. The tensor-field $X\otimes Y$ is zero.

Thus, taking any vector fields $X, Y$ on $M$ not satisfying (3) (such vector fields exist on every nonempty manifold) one obtains an example where the composition of two vector fields is not a vector field.

The missing assumption in your homework is:

There exists a point $p\in M$ such that $X(p)\ne 0$ and $Y(p)\ne 0$.