5
$\begingroup$

Let $M$ and $N$ be smooth manifods and $f:M\to N$ a smooth submersion. Prove that for every vector field $X\in \mathfrak{X}(N)$ there is a vector field $Y\in \mathfrak{X}(M)$ such that $X$ and $Y$ are $f$-related (i.e., $f_{*}Y=X\circ f $).

Here is where I'm at: take $p\in M$, then $X(f(p))\in T_{f(p)}N$. Since $f$ is a submersion, $\exists v\in T_pM$ such that $f_{*_{p}}(v)=X(f(p))$. In that fashion, we can define a function $Y:M\to TM$ with $Y(p)=v$.

I suppose this is the natural way to start. The problem is that this $v$ is not uniquely defined, which means $Y$ may well not be smooth depending on the choices for $v$, so I'm stuck here.

Any suggestions? Thanks!

$\endgroup$
1
  • 1
    $\begingroup$ Can you answer the question in the case where $f : \mathbb R^m \to \mathbb R^n$ is in local normal form? Once you have this you should be able to patch together a global solution using paracompactness and partitions of unity. $\endgroup$ Oct 13, 2016 at 23:47

1 Answer 1

6
$\begingroup$

Let $p \in M$. Due to the normal form of smooth submersions, there are smooth chart $(U_p, y_p)$ of $M$, $(V_{f(p)}, x_{f(p)})$ of $N$, containing $p$ and $f(p)$, respectively, such that:

$$ x_{f(p)}\circ f \circ (y_p)^{-1}(z_1, \ldots, z_m) = (z_1, \ldots, z_n)$$

Then, it isn't hard to prove that for $i \in \left \{ 1, \ldots, n\right \}$ $$\mathrm{d}f_{p}\left ( \dfrac{\partial }{\partial y^i}\bigg|_{p} \right ) = \dfrac{\partial }{\partial x^i}\bigg|_{f(p)}$$

I'm not going to write down the subindex $p$ on the components of the charts, to not oversaturate notation. On the other hand, since $X$ is a vector field, it has a representation in $V_{f(p)}$ of the form

$$ X = \sum\limits_{i=1}^{n}g_i^p \dfrac{\partial }{\partial x^i}\bigg|_{f(p)}$$

Afterwards, just define the vector field $Y^p:U_p \to TM$ by
$$Y^p = \sum\limits_{i=1}^{n}\left ( g_i^p \circ f\right ) \dfrac{\partial }{\partial y^i}\bigg|_{p}$$

It is straightforward that $Y^p$ is a smooth vector field and $$\mathrm{d}f_p\left ( Y^p(q)\right ) = X(f(q)) \quad \forall q \in U_p$$

Therefore, we have constructed a family of local vector fields that satisfies the required condition. Now, consider a partition of unity $\left \{ \xi_p\right \}_{p \in M}$ subordinate to $\left \{ U_p\right\}_{p \in M}$ and define $$ Y = \sum\limits_{p \in M}\xi_p Y^p $$

Finally, $Y$ is a global smooth vector field that is $f$-related to $X$.

$\endgroup$
0

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.