Let $n \geq 1$ and $F:N^n \rightarrow M^{n+l}$ be a smooth somewhere immersed map between smooth closed manifolds (assume $N$ is connected too). Somewhere immersed means that there exists a point $p \in N$ such that $d_p F$ has rank $n$. I am interested in the image of the set of points $p \in N$ such that $\text{rank}(d_pF)<n$.

Suppose that $\Sigma^s \subset N^n$ is a submanifold such that $\forall p \in \Sigma$ we have $\text{rank}(d_pF)=n-k$ for some $1 \leq k \leq n-1$. If $s=\text{dim} \Sigma < n-k$ then does the restriction of $F$ to $\Sigma$ define a submanifold in $M$?

The obvious thing to do is compose with the inclusion map but I do not have a clear idea of how to see that the rank of this composition is $s$?

  • $\begingroup$ What do you mean by "somewhere immersed"? $\endgroup$ – Najib Idrissi Apr 5 '18 at 10:18
  • $\begingroup$ It means that it is an immersion for at least one point. I have edited the question to include the definition. $\endgroup$ – ben Apr 5 '18 at 15:17
  • $\begingroup$ Also I don't think that the assumption that map is somewhere immersed is particularly relevant, it is just an additional property I have in the context I am interested in $\endgroup$ – ben Apr 5 '18 at 15:34

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