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Suppose we have two smooth manifolds $M_1$ and $M_2$ and a smooth map $i:M_1 \rightarrow M_2$ that is an embedding of $M_1$ into $M_2$. Moreover we have another submanifold $N \subset M_2$ that has a non empty intersection with the embedding $i(M_1)$. Then,in what situation is the preimage set $i^{-1}(N)$ a submanifold of $M_1$?

Or in other words, what do we have to assume so that the preimage set is a submanifold?

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    Do you mean "what do we have to assume **so** that the preimage set is a submanifold"? I assume so, but would just like to check.2011-12-27
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    Yes. I changed it.2011-12-27

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One hypothesis that works is that $N$ is transverse to the image of $i$, but this is not a necessary condition. For example, the preimage under the inclusion of the $x$ axis into $\mathbb R^2$ of the parabola $\{(t,t^2):t\in\mathbb R\}$ is a submanifold...

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    Unfortunately the nice transversal-property is not applicable in my situation. (That is why I'm asking.) But I guess there is no other general condition and I have to do it "by hand".2011-12-27
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    @Mark, you should probably explain what your situation is :)2011-12-27
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    That will not work.2011-12-27
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    So instead, the idea is to have people shot in the dark until they hit the hidden target. Now *that* will work!2011-12-27
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    The manifolds I have in mind are images of Weil functors and hence are fibered, too.2011-12-27