Manifolds with Boundary Collars and Orientations

Their difference vanishes on the relative interior, which is dense in the relative set. Every Euclidean derivative of the difference vanishes there by ordinary differentiation.

Those derivatives are continuous, so they also vanish at each face point. Hence the derivative of a half-space-smooth map is independent of its chosen extension.

By [L1], choose Euclidean extensions near $p$ and $f(p)$ and shrink the first neighbourhood so that its image lies in the domain of the second extension. Their ordinary composite then extends $g\circ f$ near $p$.

Applying [L3] to the extensions from step 1.1 gives the displayed formula, and [L2] makes the resulting derivatives independent of both extension choices. Hence $g\circ f$ is smooth and the formula is intrinsic.

On every common domain, the two coordinate representatives differ by composition on the left and right with the source and target transition maps, which are smooth by [L1].

By [L2], one representative is extension-smooth exactly when the other is, because the transition maps are diffeomorphisms with smooth inverses. By [L3], all derivatives obtained from different extensions agree on the half-space. Thus neither the charts nor the extensions affect the definition.

If $n=0$, [L1] makes the claim immediate. Assume $n\ge 1$. At any $p\in U$, choose Euclidean extensions of $f$ and $f^{-1}$ near $p$ and $f(p)$. Since both half-space composites are the identity, [L2] gives $D(f^{-1}{)}_{f(p)}D{f}_p=I$ and $D{f}_{p}D(f^{-1}{)}_{f(p)}=I$.

For $n\ge 1$, if a face point $p$ mapped to the relative interior, then on a Euclidean neighbourhood of $f(p)$ contained in $V$, the last coordinate of an extension of $f^{-1}$ would be nonnegative and would vanish at the interior point $f(p)$. Its differential there would therefore be zero, contradicting the invertibility from step 1.1. Applying the same argument to $f^{-1}$ proves the converse. Hence face and relative-interior points are preserved in every dimension, making the manifold boundary and interior intrinsic.

By [L2], every coordinate representative of $F$ and its inverse in boundary charts is a smooth half-space diffeomorphism.

Applying [L1] to those representatives shows that $F$ maps boundary points exactly to boundary points and interior points exactly to interior points. Bijectivity then gives $F(\mathrm{\partial }M)=\mathrm{\partial }N$ and $F(\mathrm{Int}M)=\mathrm{Int}N$.

If $n=0$, [L1] gives $\mathrm{Int}M=M$; it is open, and its original charts have image in ${\mathbb{R}}^0$, so the conclusion follows. Assume $n\ge 1$. Restrict each boundary chart to the inverse image of $\{x^n>0\}$. These sets are open and, by [L1], cover exactly $\mathrm{Int}M$.

In the $n\ge 1$ case, their images are Euclidean-open, and [L2] shows that the old transition maps restrict to ordinary smooth transitions. The restricted charts therefore make $\mathrm{Int}M$ a smooth boundaryless $n$-manifold; step 1.1 already handled $n=0$.

Suppose $n\ge 1$. Restrict each boundary chart to the face $x^n=0$. By [L1] these restrictions cover exactly $\mathrm{\partial }M$. Their images are open subsets of ${\mathbb{R}}^{n-1}$, and [L2] makes their transition maps smooth restrictions of extensions of the ambient transitions. They therefore define a smooth boundaryless $(n-1)$-manifold structure on $\mathrm{\partial }M$.

In a boundary chart the inclusion $\mathrm{\partial }M\hookrightarrow M$ is the coordinate map $(x^1,\dots ,x^{n-1})\mapsto (x^1,\dots ,x^{n-1},0)$, so it is a smooth injective immersion and a homeomorphism onto its subspace image. Thus it is a smooth embedding, and [L3] gives the asserted embedded submanifold. The complement is locally $\{x^n>0\}$, hence open, so $\mathrm{\partial }M$ is closed. When $n=0$, the boundary is empty by the stated convention.

If $\mathrm{\partial }M=\mathrm{\varnothing }$ and $n=0$, every boundary-chart image lies in ${\mathbb{R}}^0$ and is already Euclidean-open. If $n>0$, [L1] and [L2] show that every point has a boundary chart whose image lies in $\{x^n>0\}$ after restricting its domain. Such images are Euclidean-open. The transition maps are restrictions of the original smooth half-space transitions; on these Euclidean-open images their local Euclidean extensions show that they and their inverses are ordinary smooth maps. Hence the restricted charts form the atlas in [L3].

Conversely, suppose a covering by boundary charts has images avoiding the model face. For $n>0$, [L1] makes every covered point interior, hence $\mathrm{\partial }M=\mathrm{\varnothing }$; for $n=0$, [L1] gives the same conclusion directly. Thus both implications hold.

Choose a boundary chart $\phi :U\to V$ at $p$ and put $c=\phi (p)$. The set $\phi (U\cap O)$ is relatively open in ${\mathbb{H}}^n$. For $n>0$, choose $r>0$ so small that the relative closed ball $\overline{B(c,r)}\cap {\mathbb{H}}^n$ lies in $\phi (U\cap O)$. If $c$ is interior, shrink $r$ further so $\overline{B(c,r)}\subseteq \mathrm{int}{\mathbb{H}}^n$. If $c$ lies on the boundary, $B(c,r)\cap {\mathbb{H}}^n$ is a relative half-ball. For $n=0$, use the one-point chart image.

Let $B$ be the inverse image of the relative open ball from step 1.1, and let $C$ be the inverse image of its relative closed ball. By [F2], the relative closed ball is compact; the inverse chart is continuous, so [F3] makes its image $C$ compact in $M$. As $M$ is Hausdorff, [F3] also makes $C$ closed there. Thus $p\in B\subseteq \bar{B}\subseteq C\subseteq O$, and $\bar{B}$ is a closed subset of compact $C$, hence compact by [F3]. In dimension zero the singleton chart image gives the same conclusion.

Step 2.1 works for every point and every open neighbourhood, so these relatively compact coordinate balls and half-balls form a basis.

Put $H_r={\bigcup }_{i=1}^r\overline{U_i}$ for $r\ge 1$. Every $H_r$ is compact: an open cover of this finite union restricts to an open cover of each compact $\overline{U_i}$, and the union of finitely many finite subcovers is finite. Also $U_r\subseteq \mathrm{int}{H}_r$, so the interiors of the $H_r$ cover $M$.

Set $r_1=1$. Given $r_m$, compactness of $H_{r_m}$ and the increasing open cover $(\mathrm{int}{H}_r{)}_{r\ge 1}$ give an integer $r_{m+1}>r_m$ with $H_{r_m}\subseteq \mathrm{int}{H}_{r_{m+1}}$; take the least such integer, so this recursion uses no additional choice. Put $K_m=H_{r_m}$ and $K_{-1}=K_0=\mathrm{\varnothing }$. Then $K_m\subseteq \mathrm{int}{K}_{m+1}$ and the interiors of the $K_m$ cover $M$.

Let $A_1=K_1$ and $A_m=K_m\setminus \mathrm{int}{K}_{m-1}$ for $m\ge 2$. Each $A_m$ is compact by [L3]. For each $m$, consider every tuple $(n,W,V)$ with $V$ a coordinate ball or half-ball, $W$ open, and $\bar{W}\subseteq V\subseteq \bar{V}\subseteq U_n\cap \mathrm{int}{K}_{m+1}\setminus K_{m-2}\mathrm{.}$ Their $W$-sets cover $A_m$: given $x\in A_m$, the displayed open neighbourhood contains $x$ for some $n$; use [L1] to take a relatively compact coordinate ball or half-ball $V$ with compact closure inside it, then apply [L1] again inside $V$ to take a ball or half-ball $W$ with $x\in W\subseteq \bar{W}\subseteq V$. This proves pointwise existence without selecting witnesses at every point. Compactness makes the set of finite ordered tuple lists whose $W$-sets cover $A_m$ nonempty; for an empty annulus, the empty list is eligible.

By [A2], choose one eligible finite ordered list for each $m$. If its length is $s_m$, index its tuples by pairs $(m,j)$ with $1\le j\le s_m$; thus $I=\{(m,j):m\ge 1,1\le j\le s_m\}\subseteq {\mathbb{N}}^2$ is at most countable by [L4], and is empty when all lists are empty. Write these tuples as $(n(k),W_k,V_k)$. Since the annuli cover $M$, so do the $W_k$. For $y\in M$, choose $m$ with $y\in \mathrm{int}{K}_m$. Any $V_k$ from annulus $A_j$ with $j\ge m+3$ misses this neighbourhood because it lies outside $K_{j-2}\supseteq K_m$. Only finitely many tuples come from the finitely many earlier lists. Thus $(V_k)$ is locally finite.

The constructed $W_k,V_k$ have the required nesting and cover, and $V_k$ is subordinate to an original $U_{n(k)}$ by its defining tuple.

Form the set of all pairs $(B,j)$ where $j$ indexes a member $U_j$ of the given cover, $B$ is a coordinate ball or half-ball with compact closure, and $\bar{B}\subseteq U_j$. By [L1] the balls in these pairs cover $M$, without choosing one pair for every point. By [L2], their projection to balls has an at-most-countable subcover $(B_n)$. Use ${\mathrm{A}\mathrm{C}}_{\omega }$ to choose for each listed $B_n$ an index $j(n)$ witnessing $\overline{B_n}\subseteq U_{j(n)}$. If $M=\mathrm{\varnothing }$, the empty partition already satisfies [L6]; henceforth assume it is nonempty, repeating one selected pair if the subcover is finite.

Apply [L3] to $(B_n)$, obtaining an at-most-countable family $(W_k,V_k{)}_{k\in I}$ with ${\bigcup }_k{W}_k=M$, $\overline{W_k}\subseteq V_k\subseteq \overline{V_k}\subseteq B_{n(k)}$, and $(V_k)$ locally finite. Assign each $k$ to the original cover index $j(n(k))$ from step 1.1.

Fix $k$. A chart for $V_k$ identifies the compact set $\overline{W_k}\subseteq V_k$ with a compact subset $K_k$ of a relatively open set $Q_k\subseteq {\mathbb{H}}^n$. Choose an open $O_k\subseteq {\mathbb{R}}^n$ with $O_k\cap {\mathbb{H}}^n=Q_k$. For $n>0$, [L4] gives a smooth Euclidean bump equal to one near $K_k$ and supported inside $O_k$. Its support is compact by the finite-ball construction in [L4], so its intersection with the closed half-space is compact and lies inside $Q_k$. Restrict the bump to ${\mathbb{H}}^n$, pull it back through the chart, and extend by zero outside $V_k$. Its support is a compact subset of $V_k$, so this extension is smooth both at the manifold boundary and across the edge of $V_k$ by [L5]. It is a $[0,1]$-valued smooth function $b_k$, equal to one on $\overline{W_k}$ and supported inside $V_k$. For $n=0$, use the constant-one function on the singleton chart $V_k$, extended by zero outside it.

The nonempty set of valid bumps in step 3.1 is specified independently for each $k$ in an at-most-countable family. Use ${\mathrm{A}\mathrm{C}}_{\omega }$ once to select all $b_k$ (finite choice if $I$ is finite). Because $\mathrm{supp}{b}_k\subseteq V_k$ and $(V_k)$ is locally finite, $a:={\sum }_{k\in I}{b}_k$ is locally a finite smooth sum. It is positive everywhere because the $W_k$ cover $M$ and $b_k=1$ on $W_k$. By [L5], $1\mathrm{/}a$ is smooth in boundary charts: near each point a smooth local extension of $a$ stays positive after shrinking the chart, so the ordinary reciprocal composes smoothly. Set $c_k=b_k\mathrm{/}a$.

For each original cover index $j$, put ${\varphi }_j={\sum }_{k:j(n(k))=j}{c}_k$, taking the zero function when there are no such $k$. The family is locally finite because the $V_k$ are, and every ${\varphi }_j$ is smooth and nonnegative. The union of the locally finite closed supports assigned to $j$ is closed and lies in $U_j$, so $\mathrm{supp}{\varphi }_j\subseteq U_j$. Also ${\sum }_j{\varphi }_j={\sum }_k{c}_k=1$, and each ${\varphi }_j\le 1$. By [L6], this is a smooth partition subordinate to the original cover. The empty case was handled in step 1.1.

If $n=0$, every smooth germ is constant, so [L2] makes every derivation zero and the empty coordinate family is a basis. Assume $n\ge 1$. For a boundary germ, define ${\mathrm{\partial }}_i{\mathrm{\mid }}_p$ by differentiating any smooth Euclidean extension in the $i$th coordinate. By [L1] this is well defined; linearity and the Euclidean product rule make it a derivation.

Let $v$ be any derivation and let $F$ extend a representative of a boundary germ $f$ near the coordinate point $a$. By [L2], write $F(x)-F(a)={\sum }_i(x^i-a^i)g_i(x)$ with $g_i(a)={\mathrm{\partial }}_{i}F(a)$. Restricting to the half-space and applying $v$, using [L2] and the Leibniz rule, gives $v(f)={\sum }_{i}v(x^i){\mathrm{\partial }}_i{\mathrm{\mid }}_p(f)$. Thus the coordinate derivations span. Applying a linear relation among them to each coordinate germ proves independence, so $\dim T_{p}M=n$.

By [L4], differentiating a boundary-chart transition and its inverse gives mutually inverse matrices; [L1] makes these derivatives extension independent, and their entries vary smoothly. They are the tangent transition maps. By [L3], the dual inverse matrices are the cotangent transition maps. Hence the usual tangent and cotangent bundles extend smoothly over all of $M$, including the $n=0$ case with empty matrices.

By [L1], a restricted face chart on $\mathrm{\partial }M$ has coordinate vectors ${\mathrm{\partial }}_1,\dots ,{\mathrm{\partial }}_{n-1}$. In the corresponding boundary chart on $M$, the inclusion is $i(x^1,\dots ,x^{n-1})=(x^1,\dots ,x^{n-1},0)$, so $d{i}_p$ sends those vectors to the first $n-1$ ambient coordinate derivations. Hence $d{i}_p$ is injective and its image is their span.

In the full basis from [L2], the image found in step 1.1 is precisely the last-coordinate-zero hyperplane, which [L3] identifies with the boundary-tangent vectors.

In the chosen boundary chart take $\rho =x^n$. It is nonnegative, vanishes precisely on the face, and has nonzero differential, so it is a boundary-defining function by [L1].

For any boundary-defining function $\rho$, its restriction to the face is zero, so $d{\rho }_p$ vanishes on the tangent hyperplane. Its derivative in the positive normal coordinate is nonnegative because $\rho \ge 0$ on the half-space and $\rho (p)=0$; by [L1] it is nonzero, hence positive. Therefore $d{\rho }_p(v)$ has the sign of the last coordinate of $v$, and [L2] gives $d{\rho }_p(v)>0$ exactly for inward $v$.

Choose boundary-chart neighbourhoods covering $\mathrm{\partial }M$ and, on each one, take the coordinate field with positive last component. Add $\mathrm{Int}M$ to this cover and choose a smooth partition of unity subordinate to it.

Extend each partition-weighted coordinate field by zero outside its chart and sum the locally finite family on a neighbourhood of $\mathrm{\partial }M$; the term supported in $\mathrm{Int}M$ vanishes there. At a boundary point its normal component is a positive weighted sum of positive numbers, hence is positive. Thus this directly constructed smooth field is inward everywhere on the boundary.

Fix a boundary point and extend the coordinate components of $X$ smoothly across the face of a boundary chart. By [L1], the extended Euclidean field has a unique two-sided local flow near that point.

By [L3], the extended field is tangent to the face along the face. Applying [L2] in the Euclidean chart shows that its flow preserves the face. Each sufficiently small time slice is a local diffeomorphism with inverse the negative-time slice, so an interior point cannot cross the invariant face without violating injectivity. After shrinking the flow domain, it therefore preserves the half-space and restricts to a two-sided local flow on $M$ preserving $\mathrm{\partial }M$.

Extend the coordinate components of $X$ across the face of a boundary chart at $p$. By [L2] the extension has a Euclidean local flow, and by [L1] its last component is positive at $p$.

By continuity, shrink to an ambient coordinate neighbourhood $W$ on which the last component of the extended field is at least some $c>0$, and shrink the initial neighbourhood and time so that all relevant trajectories remain in $W$. Let $h(t)$ be the last coordinate of a forward trajectory with $h(0)\ge 0$. If $h(t_1)<0$ for some $t_1>0$, let $\tau$ be the largest zero of $h$ in $[0,t_1]$; it exists by continuity. Then $h<0$ on $(\tau ,t_1]$, while $h^{\mathrm{\prime }}(t)\ge c$ there, so the one-variable mean-value theorem gives $h(t_1)-h(\tau )>0$, contradicting $h(t_1)<0=h(\tau )$. Thus the restricted forward flow remains in $M$; no analogous negative-time conclusion follows.

If $\mathrm{\partial }M=\mathrm{\varnothing }$, the empty map from $\mathrm{\varnothing }\times [0,1)$ is a collar. Otherwise take $X$ on $U$ from [F1]. Write $F(p,t)={\mathrm{\Phi }}_t(p)$ wherever the local forward solution remains in $U\cap M$. In a boundary chart, extend $X$ smoothly across the face to apply the ordinary local-flow theorem. At $(p,0)$, the differential of $(y,t)\mapsto F(y,t)$ has the $n-1$ face directions and the strictly inward vector $X_p$ as its columns, so it is invertible. The Euclidean inverse function theorem gives a local diffeomorphism of ambient chart neighbourhoods. Since the boundary coordinate of $F(y,t)$ vanishes for $t=0$ and has positive $t$-derivative near $(p,0)$, it has the sign of $t$ there. Thus the restriction to $t\ge 0$ is a smooth local collar with image relatively open in $M$. [F1, F2, The Euclidean inverse function theorem]

No positive-time trajectory in $U\cap M$ starting on $\mathrm{\partial }M$ can meet $\mathrm{\partial }M$ again. If $F(p,\tau )=q\in \mathrm{\partial }M$ for $\tau >0$, choose a boundary defining coordinate $r\ge 0$ near $q$. Strict inwardness gives $dr(X)>0$ on a smaller neighbourhood. For all sufficiently small $h>0$, the trajectory on $[\tau -h,\tau ]$ lies there, so the one-variable mean-value theorem gives $r(F(p,\tau -h))<r(q)=0$, a contradiction.

The flow map is injective wherever these forward trajectories are defined in $U\cap M$. If $F(p,t)=F(q,s)$ with $t\ge s$, uniqueness applied backwards along the common terminal segment of length $s$ gives $F(p,t-s)=q$. If $t>s$, this contradicts step 1.2. If $t=s$, uniqueness back to time zero gives $p=q$. This argument uses only uniqueness along already existing trajectories; it does not claim a negative-time flow inside $M$.

For $n\ge 1$, let $V_n\subseteq \mathrm{\partial }M$ be the union of all relatively open boundary sets on which the smooth forward flow exists, stays in $U\cap M$, and is smooth for times in $[0,2\mathrm{/}n]$. Smooth dependence and local existence in [F2] make $V_n$ open; as $n$ grows, the $V_n$ cover $\mathrm{\partial }M$. Apply [F3] to this countable cover, obtaining a smooth locally finite partition $({\varphi }_n{)}_{n\ge 1}$ with $\mathrm{supp}{\varphi }_n\subseteq V_n$. Set $\delta (p)={\sum }_{n\ge 1}{\varphi }_n(p)\mathrm{/}(2n)$. It is positive and smooth. At each $p$ only finitely many terms are nonzero. If $n_0$ is the least active index, then $p\in V_{n_0}$ and $0<\delta (p)\le 1\mathrm{/}(2n_0)<2\mathrm{/}{n}_0$, so $F(p,t)$ exists in $U\cap M$ for every $0\le t<\delta (p)$. No selection from an uncountable family of local widths is used.

Put $D=\{(p,t):p\in \mathrm{\partial }M,0\le t<\delta (p)\}$. The map $F:D\to M$ is smooth and injective by steps 2.1–2.2. It is a local diffeomorphism at $t=0$ by step 1.1. At $t>0$, follow a sufficiently short initial part of the trajectory from a local collar chart in step 1.1, then use the ordinary local flow near the remaining interior trajectory to transport that chart; uniqueness makes the composition equal to $F$ near $(p,t)$. Hence $F$ is a local diffeomorphism there too. Its image is relatively open in $M$ and contains $\mathrm{\partial }M$; the local inverses agree by injectivity, so $F$ is a smooth embedding onto this open neighbourhood.

The map $(p,s)\mapsto (p,s\delta (p))$ is a smooth diffeomorphism from $\mathrm{\partial }M\times [0,1)$ onto $D$, with inverse $(p,t)\mapsto (p,t\mathrm{/}\delta (p))$. Composing it with $F$ gives a smooth embedding $c(p,s)={\mathrm{\Phi }}_{s\delta (p)}(p)$ with open image and $c(p,0)=p$. By Smooth collars of a manifold boundary, this is a smooth collar.

Write $B=\mathrm{\partial }M$. A boundary chart shrunk to a product $V\times [0,a)$ can be used on one copy and reflected on the other. The quotient neighbourhood is then $V\times (-a,a)$, with the two halves given by the signs of the last coordinate. Each restriction is smooth in the original half-space calculus; this supplies the asserted local model without a global collar. If $B=\mathrm{\varnothing }$, including $n=0$, the double is simply the disjoint union of two boundaryless copies and all assertions follow directly. Henceforth assume $B\mathrm{\ne }\mathrm{\varnothing }$ and ${\mathrm{A}\mathrm{C}}_{\omega }$.

Choose a collar $c:B\times [0,a)\to M$ by [F2]. On a boundary coordinate patch $y:V\to {\mathbb{R}}^{n-1}$ define the inverse seam chart by $S_c(p,t)=[c(p,t),+]$ for $t\ge 0$ and $S_c(p,t)=[c(p,-t),-]$ for $t\le 0$, using the seam identification at zero; its coordinates are $(y(p),t)$. Between two such charts for this fixed collar the transition is $(y,t)\mapsto (\tilde{y}\circ y^{-1}(y),t)$. Overlaps with interior charts lie in $t>0$ or $t<0$, where the collar and reflection are smooth diffeomorphisms. Thus these charts form a smooth atlas.

This atlas has the quotient topology. The folding map $DM\to M$ is continuous, so points with different images have disjoint open neighbourhoods. Two distinct points with the same image lie in opposite interiors, which are disjoint open sets. Hence $DM$ is Hausdorff. A countable base of $M$ gives a countable base of $DM$: use the symmetric images of a base open set in both copies, and the base open sets restricted to either interior. The symmetric sets suffice at seam points by intersecting the two preimage neighbourhoods in $M$. Thus $DM$ is second countable. Denote the resulting boundaryless manifold for a collar $c_0$ by $D_0$; each labelled copy is a closed smooth submanifold with boundary of $D_0$.

Let $c_0,c_1$ be the two collars to compare. On the intersection of their images set $X_i=(c_i{)}_{\ast }{\mathrm{\partial }}_t$. These are smooth inward fields. Their coordinate components extend locally across $B$ in $D_0$ by the definition of half-space smoothness, and a partition of unity [F5] glues the extensions on a neighbourhood of $B$, retaining their values on the positive copy. Choose a smooth function $\theta :\mathbb{R}\to [0,1]$ equal to zero near $(-\mathrm{\infty },0]$ and one near $[1,\mathrm{\infty })$, and put $X_s=(1-\theta (s))X_0+\theta (s)X_1$. In the signed $c_0$ coordinate $r$, both $dr(X_i)>0$ on $B$. Shrink the neighbourhood so both remain positive there. Then every $X_s$ is transverse inward there.

Let $C_s(p,t)$ be the flow of $X_s$ from $p\in B$ for sufficiently small $t\ge 0$, restricting to flow segments that stay in that neighbourhood. This is smooth jointly by [F4], using local coordinates, and $C_s(p,0)=p$. Its differential at $t=0$ sends $(v,b)$ to $v+b{X}_s(p)$ and is invertible. Smooth extension and [F3] give local inverses, also jointly with $s$. For fixed $s$, two such flow segments cannot meet with different initial data: uniqueness would place them on the same trajectory, which cannot cross $r=0$ twice because $dr(X_s)>0$. Thus, after restricting to the open domain where the differential is invertible, $(s,p,t)\mapsto (s,C_s(p,t))$ is a diffeomorphism onto a relative neighbourhood of $[0,1]\times B$. The allowed widths can depend on $p$; compactness of $[0,1]$ supplies a common positive width locally near each $p$. At the endpoints, uniqueness gives $C_i(p,t)=c_i(p,t)$ for sufficiently small $t$, since $X_i$ is exactly the collar velocity field.

On this neighbourhood in $\mathbb{R}\times M$ define $V_s(C_s(p,t))={\mathrm{\partial }}_s{C}_s(p,t)$. This is smooth and $V_s(p)=0$ for $p\in B$. Extend its coordinate components smoothly across the seam and glue by [F5] on $\mathbb{R}\times D_0$. The resulting field ${\tilde{V}}_s$ agrees with $V_s$ on a possibly smaller positive-side neighbourhood of $[0,1]\times B$ and vanishes on that seam. This construction extends a vector field, whose values can be added in each tangent space; it does not average manifold-valued maps.

Choose a smooth proper $h:D_0\to [0,\mathrm{\infty })$ by [F7]. Since ${\tilde{V}}_s=0$ along the seam, there is an open neighbourhood $O$ of the closed set $A=[0,1]\times B$ in $\mathbb{R}\times D_0$ where the extension is defined and $\mathrm{\mid }dh({\tilde{V}}_s)\mathrm{\mid }<1$. By [F6] choose $0\le \chi \le 1$, equal to one near $A$, with support in $O$. Extend $Z_s=\chi (s,\cdot ){\tilde{V}}_s$ by zero outside $O$. It is globally smooth, vanishes on $B$ for $0\le s\le 1$, agrees with $V_s$ near $A$ on the positive side, and satisfies $\mathrm{\mid }dh(Z_s)\mathrm{\mid }\le 1$.