What are some "interesting" examples of submersions that are not surjective?
Usually, the notion of submersion comes with a prefix of surjective. Most of the maps we come across when we do differential geometry are surjective submersions.
It arises a question, are these two properties necessarily be combined always?
One might wonder if there are surjective maps that are not submersions. There are many such maps, but one that immediately comes to mind is the map $f:\mathbb{R}\rightarrow \mathbb{R}$ defined as $x\mapsto x^3$.
Next question is, are there submersions that are not surjective? Quick checking of "some" examples discussed in a first course tells that, when ever we talk about submersion, the property of surjectivity comes with it. It does not mean there are none.
Consider a manifold $M$ and an open subset $U$ of $M$. Consider the inclusion map $i:U\rightarrow M$. It is easy to see that this map is a submersion. But, if $U$ is a proper open subset then, this inclusion map is not a surjective map. Thus, there are submersions that are not surjective.
What are some "interesting" examples of submersions that are not surjective which are not of the form $U\rightarrow M$ for an open subset $U$ of $M$?
Please do not assume the reader knows any theorems other than the definition of surjective maps, submersions.
Best Answer
It is an exercise (following from the fact that linear projections are open maps) that all submersions are open maps. If $f\colon X\to Y$ is a submersion and $X$ is compact, then $f(X)$ will be both open and closed in $Y$, hence a connected component of $Y$. (So, for an example, take $X$ to be any compact manifold, and let $Y$ be two disjoint copies of $X$; let $f$ map to the first copy by the identity map. I could make slightly more interesting examples by taking cylinders for $X$, mapping to the base of the cylinder ... or $S^3\to S^2$ by the Hopf map, then map to $S^2\times S^2$ by mapping only to the first factor, etc.)
Now, if $X$ is not compact, you already said that you could give examples by including open subsets ...