Let $M$ be a compact smooth manifold (closed, for simplicity), $n\in\mathbb{N}$ and equip the space of embeddings $Emb(M,\mathbb{R^n})$ wih the Whitney-$C^{\infty}$-topology. (The weak and the strong one are the same as $M$ is assumed to be compact.) Fix an embedding $j\in Emb(M,\mathbb{R}^n)$ and a tubular neighbourhood of the embedded $j(M)\subseteq\mathbb{R}^n$.



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> Are small normal deformations of $j$ open in $Emb(M,\mathbb{R}^n)$?
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To be more precise: Equip the normal sections $\Gamma(N(j(M))$ with the Whitney-$C^{\infty}$ topology and let $U\subset\Gamma(N(j(M))$ be the open (!) set of all sections $s$ such that the induced normal deformation $\{m+s(m)|m\in j(M)\}$ of $M$ lies in the tubular neighbourhood. Every such section $s\in U$ gives me an embedding $[m\mapsto j(m)+s(j(m)]\in Emb(M,\mathbb{R}^n)$. Do these embeddings form an open subset of $Emb(M,\mathbb{R}^n)$?


Some naive thoughts:


Consider a unit circle in the plane. Every open subset of the standard embedding of the circle should contain a small rotation of the circle, which does not come from a normal deformation of the circle. This suggests that the claim is false, but I believe to remember that I saw a flavor of this kind ("small normal deformations of embeddings are open") somewhere, although I cannot figure out where it was.



Are normal deformations of an embedding open in the $C^{\infty}$-space of embeddings of a compact smooth manifold`
Not to leave this question open:


The question as posed is wrong by the example I gave. However the example I gave is in some sense the maximum which can go wrong. More precisely small normal deformations as defined in the question are open if one factors out diffeomorphisms of $M$, e.g. they are open in $Emb(M,\mathbb{R}^n)/Diff(M)$ as Igor Khavkine kindly pointed out to me.


A proof of the new claim can be constructed along the lines of the proof of Theorem 44.1 in 'Kriegl, Michor: The convenient setting of global analysis', which shows that $$Diff(M)\rightarrow Emb(M,\mathbb{R}^n)\rightarrow Emb(M,\mathbb{R}^n)/Diff(M)$$ is a fiber bundle. The local sections are exactly the normal deformations I considered. I am grateful to Oldřich Spáčil who gave me the hint digging into the proof referenced.


