The Euclidean tubular neighbourhood theorem

If $S=\mathrm{\varnothing }$, take the unique function $\delta :S\to (0,\mathrm{\infty })$. Then ${\mathrm{\Omega }}_{\delta }=\mathrm{\varnothing }$, and $E$ is a diffeomorphism from the empty manifold onto the open neighbourhood $\mathrm{\varnothing }$ of $S$. Hence assume $S\mathrm{\ne }\mathrm{\varnothing }$.

For $p\in S$ and $a>0$ set $V_a(p)=\{(q,v)\in N^{\perp }S:\mathrm{\parallel }q-p\mathrm{\parallel }<a,\mathrm{\parallel }v\mathrm{\parallel }<a\},A_p=\{a\in (0,1]:E{\mathrm{\mid }}_{V_a(p)}\text{ is a diffeomorphism onto its image}\}\mathrm{.}$ The normal bundle has its induced topology in ${\mathbb{R}}^m\times {\mathbb{R}}^m$, so these open sets form a neighbourhood basis at $(p,0)$. Thus [L2] makes $A_p$ nonempty, and it is bounded above by one. Put $r(p)=\sup A_p\in (0,1]$. For every $0<a<r(p)$ some $b\in A_p$ satisfies $a<b$, and restricting its diffeomorphism to $V_a(p)$ proves $a\in A_p$. This is the same capped function as in [L3].

Take the positive smooth $\delta$ specifically constructed in [L3]. Its stated proof bound gives $\delta (p)\le r(p)\mathrm{/}4$ and its injectivity conclusion gives injectivity of $E$ on ${\mathrm{\Omega }}_{\delta }$. The construction uses compact sublevel minima of $r$ and explicit locally finite one-variable bumps composed with a smooth proper exhaustion; no manifold partition or additional radius selection is invoked here.

Local invertibility on the entire tube needs a separate argument. If $(p,v)\in {\mathrm{\Omega }}_{\delta }$, then $\mathrm{\parallel }v\mathrm{\parallel }<\delta (p)\le r(p)\mathrm{/}4<r(p)$. Choose a real $a$ strictly between $\mathrm{\parallel }v\mathrm{\parallel }$ and $r(p)$, for example their arithmetic mean. By step 2.1, $E{\mathrm{\mid }}_{V_a(p)}$ is a diffeomorphism, and $(p,v)\in V_a(p)$. Hence $E$ is a local diffeomorphism at every point of ${\mathrm{\Omega }}_{\delta }$. This follows from the quantitative radius bound and the eligible neighbourhoods, not from injectivity alone.

The set ${\mathrm{\Omega }}_{\delta }$ is open in the normal bundle: its defining inequality is the strict negativity of the continuous function $(p,v)\mapsto \mathrm{\parallel }v\mathrm{\parallel }-\delta (p)$. Here the vector coordinate, bundle projection and positive smooth $\delta$ are continuous. Thus the neighbourhoods from step 4.1 can be intersected with ${\mathrm{\Omega }}_{\delta }$ to give local diffeomorphisms for the restricted map, which is injective by step 3.1.

A local diffeomorphism is open. Hence $U:=E({\mathrm{\Omega }}_{\delta })$ is open and contains $S$ because $E(p,0)=p$. The injective local diffeomorphism $E:{\mathrm{\Omega }}_{\delta }\to U$ is a homeomorphism, and its local smooth inverses agree and assemble to a smooth global inverse. Thus $E$ is the required diffeomorphism. Together with the empty case in step 1.1, this proves the theorem.