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Let $i\colon L\to M$ be a submanifold inclusion. The tubular neighborhood theorem says that there is a tubular neighborhood of $i(L)$ in $M$ diffeomorphic to the normal bundle of $L$ in $M$, denoted by $N$, such that $i$ is the zero section.

The normal bundle has a natural projection $\pi\colon N \to M$. I am wondering if one can handle the zero section and the projection at the same time. That is, suppose we have $i\colon L\to M$ a submanifold inclusion, and a surjective submersion $p\colon M\to L$, such that $p\circ i=\mathrm{id}$. Question: Is it possible to find a tubular neighborhood, such that $i$ becomes the zero section and at the same time $p$ becomes the natural projection $\pi$?

I find that this is possible locally. One first consider the projection then make a change of coordinates to fix the zero section.

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