Suppose that $M$ is a smooth manifold and $U \subseteq M$ is an open subset that is also an embedded submanifold of $M$. Let $S \subseteq M$ is an immersed submanifold of $M$ such that $U \cap S \neq \emptyset$. Is it in general $U \cap S \subseteq U$ is an embedded or immersed submanifold of $U$ ?

$\textbf{Edit :}$

I need this result to show that whether the restriction of a smooth function $\tau : U \to N$ (where $U \subseteq M$ is the open submanifold as above and $N$ is a smooth manifold) to the intersection $U \cap S \subseteq U$ can be smooth or not. If i can show that $U \cap S$ is an embedded or immersed submanifold of $U$, then i'm done.

  • $\begingroup$ In general, $U\cap S$ need not be embedded, for example let $U=M$ and let $S$ be an immersed but not embedded manifold of $M$ (there are many such examples) $\endgroup$ – ArtW Jan 23 '18 at 15:11

$S$ is an immersed submanifold of $M$, i.e. there is a smooth manifold $E$ and an immersion $f\colon E\rightarrow M$ with $f(E)=S$. Then $E':=f^{-1}(U)\subset E$ is open. Now being an immersion is a local property, thus also $f\vert_{E'}\colon E'\rightarrow U\subset M$ is an immersion with image $S\cap U$. This shows that $S\cap U$ is an immersed submanifold of $U$.

In general there is no reason to assume that $S\cap U$ will be embedded (trivial counterexample: $U=M$).

  • $\begingroup$ My notation was a little confusing, I have fixed that. Now $E$ is the manifold that is being immersed and open subsets of manifolds (like $E'$) are trivially embedded submanifolds. $\endgroup$ – Jan Bohr Jan 23 '18 at 15:35
  • $\begingroup$ Is any immersed submanifold $S \subseteq M$ always appear as image of a smooth immersion $f : E \to M$ ? I know that $f(E) $ of a smooth immersion $f : E \to M$ is an immersed submanfold. But i dont know converse is always true. $\endgroup$ – kelvinn aja Jan 23 '18 at 15:37
  • $\begingroup$ That is my definition of an immersed submanifold (see also here). What definition are you referring to? $\endgroup$ – Jan Bohr Jan 23 '18 at 15:39
  • $\begingroup$ I'm using Lee's smooth Manifold $\endgroup$ – kelvinn aja Jan 23 '18 at 15:41
  • $\begingroup$ Lee on says on page 186: "An immersed submanifold of dimension k (or immersed k-submanifold) of $M$ is a subset $S \subset M$ endowed with a $k$-manifold topology (not necessarily the subspace topology) together with a smooth structure such that the inclusion map $S \hookrightarrow M$ is a smooth immersion." I simply use the name $E$ for the set $S$ equipped with its new topological and smooth structure. $\endgroup$ – Jan Bohr Jan 23 '18 at 15:47

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