Rank Theorems and Embedded Submanifolds

Fix an immersion point $p$. Choose charts around $p$ and $F(p)$ as in [L2], and let $f$ be the coordinate representative of $F$. Then $\mathrm{rank}Df(\phi (p))=m$ by [F1]. Since the differential of a map ${\mathbb{R}}^m\to {\mathbb{R}}^n$ cannot have rank above $m$, [L1] gives a Euclidean neighbourhood on which the rank stays at least $m$, hence exactly $m$.

The same argument with $n$ in place of $m$ shows that near any submersion point the rank stays equal to $n$, so the submersion locus is open. Here the upper bound $\mathrm{rank}Df\le n$ is the relevant maximal-rank bound.

Translating step 1.1 back through the source chart, every point of a smaller neighbourhood of $p$ is again an immersion point. Hence the immersion locus is open.

Steps 2.1 and 1.2 prove the two openness claims.

Choose smooth charts $(U_0,\phi )$ at $p$ and $(V_0,\psi )$ at $F(p)$ as in [L4], and let $f:=\psi \circ F\circ {\phi }^{-1}$. By [L4], the chart maps ${\phi }^{-1}$ and $\psi$ are diffeomorphisms, so their differentials are linear isomorphisms. Applying [L1] gives $Df(\phi (p))=d{\psi }_{F(p)}\circ d{F}_p\circ d({\phi }^{-1}{)}_{\phi (p)}\mathrm{.}$ Because $d{F}_p$ is an isomorphism by [F1], the Euclidean differential $Df(\phi (p))$ is invertible.

By [L2], after shrinking to open neighbourhoods $\tilde{U}\subseteq \phi (U_0)$ and $\tilde{V}\subseteq \psi (V_0)$ of $\phi (p)$ and $\psi (F(p))$, the restriction $f\mathrm{\mid }\tilde{U}:\tilde{U}\to \tilde{V}$ is a bijection with $C^1$ inverse. Since $f$ is smooth, [L3] upgrades that inverse to a smooth one.

Put $U:={\phi }^{-1}(\tilde{U})$ and $V:={\psi }^{-1}(\tilde{V})$. Then $F{\mathrm{\mid }}_U={\psi }^{-1}\circ (f\mathrm{\mid }\tilde{U})\circ \phi$, so $F{\mathrm{\mid }}_U:U\to V$ is bijective. Its inverse is ${\phi }^{-1}\circ (f\mathrm{\mid }\tilde{U}{)}^{-1}\circ \psi$, which is smooth by [L4] and step 2.1.

Thus $F{\mathrm{\mid }}_U$ is a bijective smooth map with smooth inverse, hence a diffeomorphism of neighbourhoods.

Choose charts $(U_0,\alpha )$ at $p$ and $(V_0,\beta )$ at $F(p)$ as in [L3], and let $f:=\beta \circ F\circ {\alpha }^{-1}$. By [L3], the maps ${\alpha }^{-1}$ and $\beta$ are diffeomorphisms, so their differentials are linear isomorphisms. Therefore [L1] gives $Df(\alpha (x))=d{\beta }_{F(x)}\circ d{F}_x\circ d({\alpha }^{-1}{)}_{\alpha (x)}$ for every $x$ near $p$, and the rank of $Df(\alpha (x))$ equals the rank of $d{F}_x$. Using [F1], after shrinking $U_0$ the map $f$ has constant rank $r$ on an open Euclidean neighbourhood of $\alpha (p)$.

After permuting source and target coordinates, use [L2] to form the explicit source map $\mathrm{\Phi }$ from the first $r$ components of the smooth map $f$ and the remaining source coordinates. Thus $\mathrm{\Phi }$ is smooth. Its local inverse is unique and is $C^k$ for every finite $k$ by [L4], hence is smooth. Put $g:=f\circ {\mathrm{\Phi }}^{-1}$. By [L5], after shrinking to a product neighbourhood, $g(u,v)=(u,h(u))$; here $h$ is smooth because $g$ is smooth. The target shear $\tilde{\beta }(z,w):=(z,w-h(z))$ and its explicit inverse $(z,w)\mapsto (z,w+h(z))$ are smooth. Therefore $\tilde{\beta }\circ f\circ {\mathrm{\Phi }}^{-1}(u,v)=(u,0)$ near the distinguished point.

Compose the original source chart with $\mathrm{\Phi }$ and the target chart with the shear: $\phi :=\mathrm{\Phi }\circ \alpha$ and $\psi :=\tilde{\beta }\circ \beta$. By [L3] these are smooth charts, and their coordinate representative for $F$ is the normal form from step 2.1.

Therefore $F$ has the claimed local slice form around $p$. When $r=0$ or $r=\min \{m,n\}$, the empty blocks in the displayed formula are interpreted in the usual way.

Because $F$ is an immersion at $p$, the linear map $d{F}_p:T_{p}M\to T_{F(p)}N$ is injective. Its domain has dimension $m$, so its rank is $m$. Therefore [L1] supplies a neighbourhood of $p$ on which $F$ has constant rank $m$.

Apply [L2] with $r=m$. The source has no normal coordinates left, so the local model is exactly $u\mapsto (u,0)$.

This is the asserted immersion normal form.

Since $F$ is a submersion at $p$, the linear map $d{F}_p:T_{p}M\to T_{F(p)}N$ is surjective. Its target has dimension $n$, so its rank is $n$. Thus [L1] gives a neighbourhood on which $F$ has constant rank $n$.

Apply [L2] with $r=n$. The target normal factor has dimension $0$, so the normal form reads $(u,v)\mapsto u$.

This is the claimed submersion normal form.

By [L1], after shrinking about $p$ and $F(p)$ there are charts in which $F$ becomes $u\mapsto (u,0)$.

The coordinate inclusion is injective, is a homeomorphism onto the slice ${\mathbb{R}}^m\times \{0\}$ with the subspace topology, and that slice is an embedded submanifold of the ambient Euclidean space. Pulling this description back through the charts gives the stated neighbourhood $U$.

The restricted map remains an immersion because it is locally identified with the coordinate inclusion.

Let $W\subseteq M$ be open and let $y\in F(W)$. Choose $x\in W$ with $F(x)=y$. By [L1], after shrinking around $x$ and $y$, the map is identified with a coordinate projection $(u,v)\mapsto u$.

Since $W$ is open and contains $x$, [L2] gives a product neighbourhood $U_1\times U_2\subseteq W$ in those coordinates. The projection sends this product neighbourhood onto the open set $U_1$. Therefore $y$ has an open neighbourhood contained in $F(W)$.

Because every point of $F(W)$ is interior, $F(W)$ is open. Thus $F$ is an open map in the sense of [F1].

Choose charts around $p$ and $F(p)$ as in [L2], and let $f$ be the coordinate representative of $F$. By [F1], $Df(\phi (p))$ has rank $r:={\mathrm{rank}}_{p}F$, and the hypothesis says nearby ranks are at most $r$.

By [L1], the locus where $Df$ has rank at least $r$ is open. Since $\phi (p)$ lies in that locus and nearby ranks are never above $r$, there is a smaller Euclidean neighbourhood on which the rank is both at least $r$ and at most $r$, hence exactly $r$.

Pulling that neighbourhood back through the source chart, $F$ has constant rank $r$ on a neighbourhood of $p$.

Let $\phi :U\to \phi (U)$ and $\psi :V\to \psi (V)$ be slice charts. By [F1], after projecting away the zero normal coordinates, the overlap transition on $S$ is $x\mapsto {\pi }_k\text{\hspace{0.17em}⁣}(\psi \circ {\phi }^{-1}(x,0))$, where ${\pi }_k$ denotes projection onto the first $k$ coordinates. Because $\psi \circ {\phi }^{-1}$ is a smooth map between open Euclidean sets and restriction to $(x,0)$ plus projection are smooth, the restricted transition maps are smooth.

The restricted charts cover $S$ because the slice charts do. Their images are open in ${\mathbb{R}}^k$: indeed $\phi (U)\cap ({\mathbb{R}}^k\times \{0\})$ equals $({\pi }_k(\phi (U)\cap ({\mathbb{R}}^k\times \{0\})),0)$, and the slice condition in [F1] says that every point of this set has an ambient product neighbourhood whose first-factor projection stays inside the image.

By [L1], each ambient chart map $\phi$ is a homeomorphism, so its restriction identifies $U\cap S$ with $\phi (U)\cap ({\mathbb{R}}^k\times \{0\})$. Therefore the restricted charts make $U\cap S$ open exactly when it is open in the subspace topology from [F2]. The atlas therefore generates precisely the subspace topology on $S$.

Steps 1.1-3.1 prove smooth compatibility and the topology claim.

By [L1], slice-chart restrictions give a smooth structure on $S$, and [L3] shows that its inclusion is a smooth embedding. Suppose $\mathcal{T}$ is another smooth $k$-manifold structure on the same set for which the inclusion $i:(S,\mathcal{T})\hookrightarrow M$ is a smooth embedding. By [F1], $i$ is a homeomorphism onto $S$ with the same subspace topology and an immersion.

Fix $p\in S$ and a slice chart $\phi :U\to {\mathbb{R}}^m$ around $p$. Choose a chart $\theta :W\to {\mathbb{R}}^k$ on $(S,\mathcal{T})$ at $p$. Since $i$ is an immersion and its image lies in the slice ${\mathbb{R}}^k\times \{0\}$ from [F2], the differential of ${\pi }_k\circ \phi \circ i\circ {\theta }^{-1}$ is an injective endomorphism of ${\mathbb{R}}^k$, hence an isomorphism. By [L2] this map is a local diffeomorphism. Thus the slice coordinates are smooth for $\mathcal{T}$, and conversely $\theta$ is smooth for the slice-chart structure.

Every chart of $\mathcal{T}$ is smoothly compatible with the restricted slice charts, so the identity map between $\mathcal{T}$ and the slice-chart structure is a diffeomorphism. Hence the two smooth structures coincide.

The inclusion $i:S\hookrightarrow M$ is injective and, by [L1], a homeomorphism of $S$ onto its image because the domain topology of $S$ is exactly the subspace topology.

In a restricted slice chart on $S$ and the ambient slice chart on $M$, the coordinate representative of $i$ is $u\mapsto (u,0)$. Its differential is the coordinate inclusion, hence injective, so $i$ is an immersion.

Steps 1.1 and 1.2 verify the clauses in [F1], so $i$ is a smooth embedding.

Fix $p\in M$. By [F1] the map $F$ is an immersion, so [L1] gives charts near $p$ and $F(p)$ in which $F$ becomes $u\mapsto (u,0)$.

In those target coordinates, the image of a small neighbourhood of $p$ is exactly the coordinate slice ${\mathbb{R}}^m\times \{0\}$. Because [F1] also says that $F$ is a homeomorphism onto its image, we may shrink the neighbourhood in the target so that no other points of $M$ map into that same slice patch. Hence the image is locally a slice chart in the sense of [F2].

Since every point of $F(M)$ has such a slice neighbourhood, $F(M)$ is an embedded $m$-submanifold. The corestriction $F:M\to F(M)$ is already a homeomorphism by [F1], and in the local coordinates of step 1.1 it is the identity on the ${\mathbb{R}}^m$ factor, so it is a diffeomorphism.

By [L1], $F$ is continuous. Since it is injective, the corestriction $f_0:M\to F(M)$ is a continuous bijection. By [L4], the subspace $F(M)\subseteq N$ is Hausdorff.

Let $C\subseteq M$ be closed. Since $M$ is compact, [L2] makes $C$ compact. To show that $f_0[C]$ is compact in $F(M)$, let $\mathcal{U}$ be an open cover of $f_0[C]$ in the subspace $F(M)$; then $\{f_0^{-1}(U):U\in \mathcal{U}\}$ is an open cover of $C$, so finitely many members cover $C$. Applying $f_0$ back shows that the same finite subfamily covers $f_0[C]$. Thus $f_0[C]$ is compact, hence closed in the Hausdorff space $F(M)$ by [L3].

Step 2.1 shows that $f_0$ is a closed bijection. Therefore for every open set $O\subseteq M$, the complement $M\setminus O$ is closed and $f_0[O]=F(M)\setminus f_0[M\setminus O]$ is open in $F(M)$. So $f_0$ is an open bijection, hence a homeomorphism. Since $F$ is an injective immersion by hypothesis, [F1] now gives that $F$ is a smooth embedding.

Assume first that $G$ is smooth. Choose a chart $\theta$ on $N$ and a slice chart $\phi$ on $M$ around $G(x)\in S$. In the corresponding restricted chart on $S$ from [L1], the representative of $G$ has values in ${\mathbb{R}}^k$, and the representative of $i\circ G$ is obtained by appending $m-k$ zero coordinates. Hence $i\circ G$ is smooth.

Conversely, assume $i\circ G$ is smooth. In a slice chart on $M$, the image of $S$ is ${\mathbb{R}}^k\times \{0\}$ by [F1]. Therefore the representative of $i\circ G$ has last $m-k$ coordinates identically zero, and its first $k$ coordinates are exactly the representative of $G$ in the restricted slice chart from [L1]. Those first $k$ coordinates are smooth, so $G$ is smooth.

Steps 1.1 and 1.2 prove both directions.

Assume $f$ is smooth. Fix $p\in S$ and choose a slice chart $\phi :U\to \mathrm{\Omega }\subseteq {\mathbb{R}}^m$ with $\phi (S\cap U)=\mathrm{\Omega }\cap ({\mathbb{R}}^k\times \{0\})$. In these coordinates, the representative of $f{\mathrm{\mid }}_{S\cap U}$ depends only on the first $k$ variables. Extend it to all of $\mathrm{\Omega }$ by ignoring the last $m-k$ coordinates. Transporting this extension back gives a smooth $\tilde{f}:U\to N$ with $\tilde{f}{\mathrm{\mid }}_{U\cap S}=f{\mathrm{\mid }}_{U\cap S}$.

Conversely, suppose every point has such an ambient extension. On each $U\cap S$, the restriction of $\tilde{f}$ to the embedded submanifold $U\cap S$ is smooth by [L1], so $f$ is smooth near every point of $S$. Smooth maps being local on the source, $f$ is smooth on all of $S$.

Therefore the ambient-extension criterion is equivalent to smoothness of $f$.

Choose a slice chart $\phi :U\to \mathrm{\Omega }\subseteq {\mathbb{R}}^k\times {\mathbb{R}}^{m-k}$ at $p$ as in [F1], and let ${\mathrm{pr}}_2:{\mathbb{R}}^k\times {\mathbb{R}}^{m-k}\to {\mathbb{R}}^{m-k}$ be the second-factor projection. Define $\mathrm{\Phi }:={\mathrm{pr}}_2\circ \phi :U\to {\mathbb{R}}^{m-k}$. Then $S\cap U={\mathrm{\Phi }}^{-1}(0)$ by the slice description.

In the same coordinates, $\mathrm{\Phi }={\mathrm{pr}}_2\circ \phi$. By [L1], the differential $d{\phi }_q$ is an isomorphism for every $q\in U$. Applying [L2] gives $d{\mathrm{\Phi }}_q=d({\mathrm{pr}}_2{)}_{\phi (q)}\circ d{\phi }_q\mathrm{.}$ The Euclidean differential $d({\mathrm{pr}}_2{)}_{\phi (q)}$ is the coordinate projection onto the normal factor, hence surjective. Therefore $d{\mathrm{\Phi }}_q$ is surjective for every $q\in U$, so $\mathrm{\Phi }$ is a submersion.

Therefore $\mathrm{\Phi }$ satisfies the definition in [F2] and is a local defining map for $S$ at $p$.

Let $p\in F^{-1}(q)$ be arbitrary. By [F1], $F$ is a submersion at $p$.

Apply [L1] at $p$. In suitable charts around $p$ and $q$, the map becomes $(u,v)\mapsto u$ on ${\mathbb{R}}^n\times {\mathbb{R}}^{m-n}$. After centering $q$, the fibre is $\{0\}\times {\mathbb{R}}^{m-n}$. Permuting the two source-coordinate blocks sends it to the standard slice ${\mathbb{R}}^{m-n}\times \{0\}$. Thus $F^{-1}(q)$ is locally an embedded submanifold.

Since every point of the fibre has such a slice neighbourhood, $F^{-1}(q)$ is an embedded submanifold. Its local model has dimension $m-n$, so by [F2] the codimension is $n$.

Because $q$ is a regular value and $p\in F^{-1}(q)$, [F1] makes $F$ a submersion at $p$. Write $m:=\dim M$, $n:=\dim N$, and $\mathrm{\ell }:=m-n$. By [L2], choose local coordinates near $p$ and $q$ in which the representative of $F$ is $(u,v)\mapsto u$ on ${\mathbb{R}}^n\times {\mathbb{R}}^{\mathrm{\ell }}$, with $p$ and $q$ sent to the origins. Then the fibre $F^{-1}(q)$ is represented by the slice $\{0\}\times {\mathbb{R}}^{\mathrm{\ell }}$. By [L1], this is the embedded-submanifold structure on the fibre near $p$, so its tangent vectors are exactly the vectors of the form $(0,w)$.

Let $\pi (u,v)=u$ be the coordinate projection. Step 1.1 makes $\psi \circ F\circ {\phi }^{-1}=\pi$ near the distinguished point. By [L3], the differentials $d{\phi }_p$ and $d{\psi }_q$ are isomorphisms, and [L4] gives $d{\psi }_q\circ d{F}_p\circ d({\phi }^{-1}{)}_{\phi (p)}=d{\pi }_{\phi (p)}\mathrm{.}$ Because $d{\pi }_{\phi (p)}$ has kernel $\{0\}\times {\mathbb{R}}^{\mathrm{\ell }}$, one gets $\ker d{F}_p=d{\phi }_p^{-1}(\{0\}\times {\mathbb{R}}^{\mathrm{\ell }})\mathrm{.}$ Step 1.1 identifies the same subspace with $T_p(F^{-1}(q))$, so $T_p(F^{-1}(q))=\ker d{F}_p$.

Therefore the intrinsic tangent space of the regular level set equals $\ker d{F}_p$.

Fix $p\in F^{-1}(S)$ and put $y:=F(p)$. By [L1], there is a neighbourhood $V$ of $y$ and a local defining submersion $\mathrm{\Phi }:V\to {\mathbb{R}}^c$ for $S$ at $y$. Shrink to an open neighbourhood $U$ of $p$ with $F(U)\subseteq V$, and set $H:=\mathrm{\Phi }\circ F:U\to {\mathbb{R}}^c$. By [L4], $H$ is smooth, and $H^{-1}(0)=U\cap F^{-1}(S)$. For every $x\in U$, [L5] gives $d{H}_x=d{\mathrm{\Phi }}_{F(x)}\circ d{F}_x$; both factors are surjective because $F$ and $\mathrm{\Phi }$ are submersions, so $d{H}_x$ is surjective. Thus $0$ is a regular value of $H$.

By [L2], $U\cap F^{-1}(S)$ is an embedded codimension-$c$ submanifold near $p$. Since $p$ was arbitrary, $F^{-1}(S)$ is embedded of codimension $c$.

Applying [L3] to the regular level set of $H$ gives $T_p(F^{-1}(S))=\ker d{H}_p$. By [L5], $\ker d{H}_p=\ker (d{\mathrm{\Phi }}_y\circ d{F}_p)=(d{F}_p{)}^{-1}(\ker d{\mathrm{\Phi }}_y)$. Applying [L3] again to the regular level set ${\mathrm{\Phi }}^{-1}(0)=V\cap S$ at $y$ gives $\ker d{\mathrm{\Phi }}_y=T_{y}S$, so $T_p(F^{-1}(S))=(d{F}_p{)}^{-1}(T_{y}S)$.

Steps 2.1 and 2.2 prove the theorem.