I'm trying to figure out the following question:
Consider a smooth submersion $$f: M^{2n} \mapsto \mathbb{R}^{n}$$
Then for $x_0 \in M$ with $f(x_0)=c_0$, we can find an open neighborhood $U_0$ of $x_0$ and an open neighborhood $B_0$ of $c_0$ s.t. there exists a diffeomorphism $\phi: B_0 \times f^{-1}(c_0) \mapsto U_0$ with
$f(\phi (c,x))=c$ and $\phi(c_0,x)=x$
i.e. we have over $U_0$, $f$ is a trivial fibration.
My idea is to use connection to ifentify nearby fibers, precisely:
- Choose a connection $\nabla$ on M, denote its exponential map as exp.
- For each $v \in \mathbb{R}^n$ as tangent vector at $c_0 \in \mathbb{R}^n$, choose a splitting $T_{x_0}M=\text{ker}(df)\bigoplus V$ and define a lifting map $L: T_{c_0}\mathbb{R}^n \mapsto V$ by reversing the isomorphism $df\mid_{V}$
- Define $\phi$ as $\phi (b,f)=\text{exp}_f(L(b-b_0))$
I have a vague idea that this should work, but I still have some confusions in the following parts:
The local diffeomorphism comes from the fact that the exponential map itself is a local diffemorphism. But to ensure this, do we have to choose the connection in a certain way? i.e. how do I make sure that the exponential map takes a vector in $\text{ker}df$ to some other point in the same fiber?
The choice of the splitting should be done by use of a metric. Is this necessary? i.e. can we choose a splitting that solely relies on the connection itself?