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Let $M,M'\subseteq \mathbb{R^n}$ two compact embedded submanifolds, which are abstractly diffeomorphic. Tangent and normal bundle of the two submanifolds inherit a metric from $\mathbb{R^n}$.

By the tubular neighbourhood theorem, there is continuous function $\delta\colon M\rightarrow (0,\infty)$, such that $$\{(m,v)\in M\times N_mM\colon |v|<\delta(m)\}\rightarrow \mathbb{R^n},(m,v)\mapsto m+v$$ is a diffeomoprhism from an open subset of the normal bundle to an open subset of $\mathbb{R^n}$. Give the sections $\Gamma(NM)$ of the normal bundle the Whitney $C^\infty$ topology. Restricting to a neighbourhood $U\subset \Gamma(NM)$ of the zero section, we get a injective map from $U$ to the set of all submanifolds in $\mathbb{R}^n$ diffeomorphic to $M$ by first taking the graph in the normal bundle and then map the submanifold to $\mathbb{R}^n$ via the above map. The same works for $M'$ for a $\delta'\colon M'\rightarrow (0,\infty)$ and an open neighbourhood of the zero section $U'\subset\Gamma(NM')$.

After possibly restricting to submanifolds of $\mathbb{R^n}$ which come both from $U$ and $U'$, we get a map $$\Gamma(NM)\supset U\rightarrow U'\subset\Gamma(NM')$$.

Is this map always continous? I think so, but could not manage to prove it explictly.


Edit: Since my question seems not to be completely clear, I will make it even more explicit.

Let's call the map of the fourth line $\alpha_M$ and the one corresponding to M' $\alpha_{M'}$.

Define $V\colon=\{(m,v)\in M\times N_mM\colon |v|<\delta(m)\}$ and $V'\colon=\{(m,v)\in M'\times N_mM'\colon |v|<\delta'(m)\}$ then $$O\colon=\{s\in\Gamma(NM)|s(M)\subset V\}$$ and $$O'\colon=\{s\in\Gamma(NM')|s(M')\subset V'\}$$ are open subets of $\Gamma(NM)$ resp. $\Gamma(NM')$.

Let $S$ be the set of all submanifolds of $\mathbb{R}^n$. By first taking the graph of the section and then applying $\alpha_M$ resp. $\alpha_{M'}$ to it, we get injective maps $$f\colon O\rightarrow S\text{ and } f'\colon O'\rightarrow S.$$ Now suppose their images have nonemtpy intersection. Then we get a map $$f^{-1}(f(O)\cap f'(O'))\rightarrow f'^{-1}(f(O)\cap f'(O'))$$ from a subset of $\Gamma(NM)$ to a subset of $\Gamma(NM')$ and my question is whether this map is always continuous.

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  • $\begingroup$ I don't see how you pass from an $M_1\subset U$ (or its section) to another $M'_1\subset U'$. You have a diffeomorphism $M\to M'$,but not between the tubes $U$ and $U'$. $\endgroup$ – Jesus RS Mar 12 '15 at 11:11
  • $\begingroup$ $U$ and $U'$ are not the tubes, they are the neighbourhoods of the zero sections of $\Gamma(NM)$ respectively $\Gamma(NM')$, such that the set of submanifolds which come from a graph of a section in $U$ and the set of submanifolds which come from one of $U'$ are equal. $\endgroup$ – Tom Mar 12 '15 at 15:40
  • $\begingroup$ That final sentence is what I do not understand. You pass from a section $\sigma$ in $U$ to its graph in the bundle, then to a submanifold $N$ in the tube $W$ of $M$. The same for $M'$ to get from some $\sigma'$ an $N'$. (I was using the same letter for $W$ and $U$.) But, how to pas from $N$ to $N'$? I don't see any diffeo from $W$ to $W'$. $\endgroup$ – Jesus RS Mar 12 '15 at 16:10
  • $\begingroup$ I added more details to the question regarding your comment. $\endgroup$ – Tom Mar 12 '15 at 16:35

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