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.

In half-space charts, restrict the Euclidean bumps used in the boundaryless construction; their supports and local finiteness survive restriction.

The same countable locally finite shrinking and normalization construction produces a positive locally finite family summing to one. Thus it is subordinate to the prescribed cover, with the same ${\mathrm{A}\mathrm{C}}_{\omega }$-sufficient choice bound as the precursor construction.

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.

Flow a global inward field from boundary points. In a boundary chart, extend the field across the face. The derivative of $(y,t)\mapsto {\mathrm{\Phi }}_t(y,0)$ at $t=0$ is the identity on face directions together with the inward vector in the time direction, hence is invertible. The Euclidean inverse function theorem therefore gives local collar embeddings.

A locally finite refinement of these local collars admits a smooth positive width $\delta$ for which $(p,t)\mapsto {\mathrm{\Phi }}_t(p)$ is injective on $0\le t<\delta (p)$ and has open image near the boundary; this is the usual locally finite shrinking of local collar domains.

The reparametrization $c(p,s)={\mathrm{\Phi }}_{s\delta (p)}(p)$ from $\mathrm{\partial }M\times [0,1)$ is then a smooth embedding, fixes $\mathrm{\partial }M$ at $s=0$, and has that open image. It is therefore 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 $\mathrm{\mid }dh(Z_s)\mathrm{\mid }\le 1$.

The local evolution of $Z_s$ from [F8] exists for the whole interval $[0,1]$ in either time direction. Indeed, along a trajectory starting at $x$, the last bound keeps $h$ at most $h(x)+1$, a compact sublevel. Cover the product of that compact sublevel and $[0,1]$ by finitely many local evolution neighbourhoods from [F8]; their smaller neighbourhoods give a positive uniform continuation time. Consequently a finite endpoint in $[0,1]$ cannot be maximal. Uniqueness makes forward and reverse evolutions inverse smooth maps. The evolution fixes $B$ pointwise. A trajectory cannot meet $B$ from outside it, since reverse uniqueness would make that trajectory constant; hence it preserves each labelled half. Its time-one restriction $H:M\to M$ on the positive copy is a boundary-fixing diffeomorphism.

For each $p\in B$, compactness of $[0,1]$ and continuity of $C_s(p,t)$ let us shrink a neighbourhood of $(p,0)$ so all the paths $s\mapsto C_s(q,t)$ stay where $Z_s=V_s$. They then solve the evolution equation with initial value $c_0(q,t)$, so uniqueness yields $H(c_0(q,t))=c_1(q,t)$ there. Define $DH:DM\to DM$ by applying this same $H$ to each labelled copy. It is well-defined and bijective, fixes the seam, and preserves labels. In the $c_0$ source and $c_1$ target seam charts it is exactly $(y,t)\mapsto (y,t)$; off the seam it and its inverse are smooth because $H$ is a diffeomorphism. Therefore $DH:D_0\to D_1$ is the required diffeomorphism.

By [L1], choose a seam chart at $p$ in which the labelled copy of $M$ is a half-space. A function extends there by [L2]. For a tensor field, [L3] expresses it in the smooth coordinate frame with finitely many smooth component functions, each of which extends by [L2].

Reassemble the extended components in the same coordinate frame and restrict to a smaller neighbourhood of $p$ in the double. The resulting tensor is smooth and restricts to $T$ on $M$. Since [L2] supplies no unique Euclidean extension, this construction is not canonical.

At an interior point, [L1] and [L3] give an ordinary slice chart. At a boundary point, choose boundary coordinates $x$ on $S$ and ordinary ambient coordinates. By [L2], extend the coordinate embedding to a smooth map $\tilde{F}$ on an open subset of ${\mathbb{R}}^k$. Its derivative at the boundary point equals the injective differential from [L1], so after shrinking an invertible $k\times k$ minor stays nonzero and $\tilde{F}$ has constant rank $k$.

Apply [L3] to $\tilde{F}$. In the resulting source coordinates $u=\alpha (x)$ and target coordinates $(u,w)$, it has the form $u\mapsto (u,0)$. The source change $\alpha$ need not preserve the face, but it can be absorbed into the target chart: postcompose that chart with the local diffeomorphism $(u,w)⟼({\alpha }^{-1}(u),w)\mathrm{.}$ In the new target coordinates, $\tilde{F}(x)=(x,0)$ in the original boundary coordinates.

Restricting $x$ back to the source half-space now gives $S=\{x^{k+1}=\cdots =x^n=0,x^k\ge 0\}$ near the boundary point, while step 1.1 gives the ordinary slice at interior points.

If $p\in \mathrm{Int}S$, then [L1] puts $p$ in $\mathrm{Int}M$, and [L2] applies inside $\mathrm{Int}M$. Now let $p\in \mathrm{\partial }S=S\cap \mathrm{\partial }M$. Choose boundary coordinates $(u,s)$ on $S$ and $(z,t)$ on $M$, and write the inclusion as $F(u,s)=(G(u,s),h(u,s))$. By [L1], $h(u,0)=0$ and transversality gives $d(h{\mathrm{\mid }}_S{)}_p\mathrm{\ne }0$. Its derivatives in the $u$-directions vanish on the face, so ${\mathrm{\partial }}_{s}h(p)\mathrm{\ne }0$; it is positive because $h(u,s)\ge 0$ for $s\ge 0$.

By [L3], replacing the source normal coordinate $s$ by $h(u,s)$ is a half-space-preserving local coordinate change. Thus assume $F(u,s)=(G(u,s),s)$. The restriction of $F$ to the face is an embedding, so $D_{u}G(u,0)$ has rank $k-1$. If $k>1$, choose an invertible $(k-1)\times (k-1)$ minor and use [L3] again in a coordinate change preserving $s$; if $k=1$, this change is empty. In either case $F(u,s)=(u,H(u,s),s)$, where $H$ has $n-k$ components.

The target coordinate change $(z^{\mathrm{\prime }},z^{\mathrm{\prime }\mathrm{\prime }},t)⟼(z^{\mathrm{\prime }},z^{\mathrm{\prime }\mathrm{\prime }}-H(z^{\mathrm{\prime }},t),t)$ is a half-space-preserving local diffeomorphism, with inverse obtained by adding $H(z^{\mathrm{\prime }},t)$. It sends $S$ to the coordinate half-slice $\{z^{\mathrm{\prime }\mathrm{\prime }}=0,t\ge 0\}$ and sends $S\cap \mathrm{\partial }M$ to its face $\{z^{\mathrm{\prime }\mathrm{\prime }}=0,t=0\}$. This proves the simultaneous straightening and the asserted equality of induced boundary structures.

If $n=0$, every differential to the zero tangent space is surjective, so both critical-value sets are empty. Suppose $n\ge 1$. By [L1] and second countability, choose countable coordinate covers of $\mathrm{Int}M$ and, when $m\ge 1$, of $\mathrm{\partial }M$, refining them so each image lies in a target chart. Each coordinate representative has a $C^r$ Euclidean extension. Since $r>\max \{m-n,0\}$ also implies $r>\max \{(m-1)-n,0\}$, [L2] makes every chartwise critical-value set null.

By [L3], each restriction-critical-value set and their union are null. A manifold null set has empty interior, so its complement is dense; that complement consists exactly of the values regular for both restrictions. Together with the $n=0$ case of step 1.1, this proves the claim.

The wedge of an ordered basis is nonzero in $\det V$, and changing basis multiplies it by the determinant of the change-of-basis matrix.

Thus two bases determine the same ray exactly when their change determinant is positive. When $n=0$, the unique empty wedge is $+1$, so it represents only the positive one of the two rays in [L1].

Suppose an orientation is selected and either $n\ge 2$, or $n>0$ and $\mathrm{\partial }M=\mathrm{\varnothing }$. By [L1] and [L2], each chart frame has a sign relative to that ray; continuity makes this sign locally constant, so restrict to its sign components. On a negative interior chart, reverse one coordinate. On a negative boundary chart with $n\ge 2$, reverse one of the first $n-1$ coordinates; this preserves ${\mathbb{H}}^n$ while reversing the frame orientation. The resulting positive charts cover $M$.

On overlaps, [L2] says that both coordinate frames are positive precisely when their change determinant is positive. Thus step 1.1 gives a positive-transition atlas. Conversely, in every $n>0$, positive transition determinants make the chart-frame rays agree on overlaps and hence define the orientation of [L1].

The converse in step 2.1 is not reversible for an arbitrary orientation when $n=1$ and there is boundary: by [L3], a positive boundary chart necessarily declares the inward vector positive. On the standard oriented interval $[0,1]$, $+{\mathrm{\partial }}_x$ is inward at $0$ but outward at $1$, so its orientation cannot be represented by positive boundary charts at both endpoints. Finally, when $n=0$, [L1] shows why independent pointwise signs, rather than the unique empty chart frame, remain the correct datum.

A nonzero top form selects the determinant ray on which it is positive, producing an orientation.

Conversely choose positive local top forms for an orientation and multiply them by a subordinate partition of unity. At every point all nonzero summands lie in the same positive ray, so their sum is nonzero and is a global top form.

By [L1], at each point any other orientation $o^{\mathrm{\prime }}$ is either $o$ or its opposite. Smoothness of both ray choices makes the relative sign locally constant.

Connectedness makes that sign constant, so $o^{\mathrm{\prime }}=o$ everywhere or $o^{\mathrm{\prime }}=-o$ everywhere. Both choices exist and are distinct because $M$ is nonempty. On a disconnected manifold the same locally constant sign may be selected independently on each component.

By [L1] and [L2], $d{F}_p$ is an isomorphism. Its determinant map therefore sends the selected source ray to exactly one of the two target rays. Choose local nonzero determinant sections ${\sigma }_M$ and ${\sigma }_N$ in the selected rays. There is a smooth nowhere-zero scalar $\lambda$ such that $\det (dF)({\sigma }_M)=\lambda {\sigma }_N$; its sign is precisely whether the selected rays are preserved or reversed.

Since $\lambda$ is continuous and never zero, its sign is locally constant and therefore constant when $M$ is nonempty and connected. If oriented source and target charts happen to be available, their coordinate determinants may be used for ${\sigma }_M$ and ${\sigma }_N$, and then $\lambda$ is the Jacobian determinant. The determinant-line formulation remains valid at one-dimensional boundary points and in dimension zero, where such chart frames need not encode every selected ray.

By [L2], the images of $n_0$ and $n_1$ in the one-dimensional quotient $T_{p}M\mathrm{/}{T}_p\mathrm{\partial }M$ lie on the same ray. Hence $n_1=a{n}_0+w$ for some $a>0$ and $w\in T_p\mathrm{\partial }M$.