Compatible tubular neighbourhoods agree near compact sets up to ambient isotopy

By [L1], after restricting their domains near the zero section the two tubular maps have a transition diffeomorphism $\mathrm{\Psi }={\mathrm{\Phi }}_2^{-1}\circ {\mathrm{\Phi }}_1$ fixing that section, with ${\mathrm{\Phi }}_2\circ \mathrm{\Psi }={\mathrm{\Phi }}_1$. Its differential is the identity on the zero-section tangent space and induces the identity on the vertical normal quotient: the latter follows by composing the equal normal identifications of $d{\mathrm{\Phi }}_1$ and $d{\mathrm{\Phi }}_2$. There is no claim that $\mathrm{\Psi }$ maps a chosen disc bundle onto itself.

For $t>0$ define ${\mathrm{\Psi }}_t={\delta }_{1\mathrm{/}t}\circ \mathrm{\Psi }\circ {\delta }_t$ wherever defined. In a bundle chart write $\mathrm{\Psi }(x,v)=(b(x,v),w(x,v))$, where the second coordinate is in the fibre over $b(x,v)$. Fixing the zero section gives $b(x,0)=x$ and $w(x,0)=0$, and the normal derivative condition gives $D_{v}w(x,0)=I$. In these charts the conjugation is $(b(x,tv),w(x,tv)\mathrm{/}t)$. The identity $\frac{w(x,tv)}{t}={\int }_0^1{D}_{v}w(x,utv)vdu$ extends smoothly to $t=0$, with value $v$; the base component extends to $x$. Thus ${\mathrm{\Psi }}_0=\mathrm{i}\mathrm{d}$ and the family is smooth up to $t=0$. Applying the same calculation to ${\mathrm{\Psi }}^{-1}$ gives a smooth inverse family near the zero section. These extensions agree on overlaps, since the positive-time formulas are intrinsic and equality extends to zero by continuity. Every ${\mathrm{\Psi }}_t$ fixes the zero section.

Shrink to a common open disc neighbourhood ${\mathrm{\Omega }}^{\mathrm{\prime }}$ on which these families are defined and ${\mathrm{\Psi }}_t({\mathrm{\Omega }}^{\mathrm{\prime }})$ lies in the domain of ${\mathrm{\Phi }}_2$ for all $0\le t\le 1$. Such a neighbourhood exists locally over every point of $S$: the maps and inverse maps in step 1.2 are defined on open neighbourhoods of the zero section times the compact parameter interval, so finitely many parameter neighbourhoods give one local fibre-radius bound. Refine the resulting base cover and use [L5] to take a positive smooth radius below the local bounds, shrinking the original disc radii as well. For each positive $t$, ${\mathrm{\Psi }}_t$ is injective on its domain since it is a conjugate of a diffeomorphism; at $t=0$ it is the identity. Its smooth inverse in step 1.2 makes each restriction an open embedding. Put ${\mathrm{\Phi }}_s={\mathrm{\Phi }}_2\circ {\mathrm{\Psi }}_{1-s}$ on ${\mathrm{\Omega }}^{\mathrm{\prime }}$. Then ${\mathrm{\Phi }}_0={\mathrm{\Phi }}_1$, ${\mathrm{\Phi }}_1={\mathrm{\Phi }}_2$, and each ${\mathrm{\Phi }}_s$ is a tubular neighbourhood embedding fixing $S$ pointwise. The level-preserving track map is an open embedding: its slice differential is invertible, its time component is the identity, and the inverse is smooth by the inverse family.

Let $U={\mathrm{\Phi }}_1({\mathrm{\Omega }}^{\mathrm{\prime }})\subseteq N$ and define $J_s={\mathrm{\Phi }}_s\circ {\mathrm{\Phi }}_1^{-1}:U\to N$. This is a smooth isotopy of the ambient open set $U$ with open track image, and $J_0$ is the inclusion. For a compact $A\subseteq S\subseteq U$, apply [L2] to obtain a compactly supported ambient isotopy $H$ agreeing with $J_s$ on a neighbourhood of $A$. Hence $H_s\circ {\mathrm{\Phi }}_1={\mathrm{\Phi }}_s$ near $A$ in the normal bundle, and $H_1\circ {\mathrm{\Phi }}_1={\mathrm{\Phi }}_2$ there. Since every ${\mathrm{\Phi }}_s$ fixes the zero section, $H_s$ fixes a neighbourhood of $A$ in $S$ pointwise.

When $S$ is compact, take $A=S$. Its zero-section track is $S\times I$, which is compact; no compactness of the entire open tubular domain is needed. Every point of this compact track lies in any prescribed open neighbourhood $W$ of ${\mathrm{\Phi }}_1(\mathrm{\Omega })\cup {\mathrm{\Phi }}_2(\mathrm{\Omega })$ times $I$. By [L4] choose the relatively compact cutoff for the open-track velocity construction of [L2] inside $W\times I$. The resulting flow is supported in a compact subset of $W$ and has the same agreement near all of $S$. This proves all conclusions with the stated normal-identification hypothesis.