# What is a submersion?

Don't misunderstand me, I read the Wikipedia article and I understood the definition - i.e. a function $$f:\mathbb R^m\supseteq U\to\mathbb R^n$$ is a submersion if $$rk(f)\equiv n$$, meaning that the differential is surjective everywhere.

I also know the connection to submanifolds, stating that a subset $$M\subseteq\mathbb R^n$$ is an $$m$$-dimensional $$C^k$$-submanifold if for every $$p\in M$$ there exists an environment $$V\subseteq\mathbb R^n$$ and a $$C^k$$-submersion $$f:V\to\mathbb R^{n-m}$$ such that $$M\cap V=f^{-1}(0)$$.

But what is a submersion? Is there any visual understanding of it? And how can one intuitively understand the connection to submanifolds?

I'm aware that this is not a mathematical precise question and hence am not expecting formally correct answers, I'm just trying to gain a deeper understanding to submersions.

• By the rank theorem, a submersion is locally a projection, so the effect of $f$ locally, is to collapse $\mathbb R^m$ to $\mathbb R^n.$ – Matematleta Jul 27 at 20:26

## 2 Answers

I think the answer you're looking for is very simple. The prototype of a submersion is the (linear) projection map $$\pi\colon \Bbb R^m\to\Bbb R^n$$ with $$m\ge n$$: Writing $$\Bbb R^m = \Bbb R^n\times\Bbb R^{m-n}$$, the map is given by $$\pi(x,y)=x$$. As an application of the inverse function theorem, you can prove that, any submersion $$f\colon X\to Y$$ ($$X$$, $$Y$$ manifolds—your $$U$$ and $$\Bbb R^n$$ if you like), in appropriate local coordinates on $$X$$ and $$Y$$ the mapping $$f$$ will be given exactly by $$\pi$$.

• This does help, but I think I'm really asking for something impossible, as I'm basically trying to get a visual understanding of something that only seems to be reasonable to even look at in more than three dimensions, which is impossible to imagine. – MetaColon Jul 27 at 22:22
• Certainly not. You can map, for example, a torus to a circle or — one of the most important maps in mathematics — the Hopf map from $S^3$ to $S^2$. (This is a fundamental example of a bundle with fibers $S^1$.) – Ted Shifrin Jul 27 at 22:34

Let me recall a famous result due to Ehresmann:

Theorem. (Ehresmann.) A proper submersion is a fiber bundle.

So, basically, if you work in a compact domain (a closed manifold), then you can visualize submersions $$f\colon M \to N$$ as maps which allow you to see the domain locally as a product of the fiber and an open set in the codomain. Think for example about the natural projection of the tangent bundle. This has lots of implications, for instance you can show that the Euler-Poincaré satisfies $$\chi(M)=\chi(N)\cdot \chi(F)$$ where $$F$$ denotes the fiber space.

• I was kind of hoping for a simple explanation - whilst the connection to fiber bundles might be an interesting (and even intuitive) one, as I know nothing about fiber bundles this doesn't quite help me that much. It seems as if I'd need a deeper understanding of topology itself to grasp the concept of fiber bundles. – MetaColon Jul 27 at 19:13