Rank of a Smooth Map
To differentiate a map is to replace it, near a point, by the linear map that best matches it. For
a smooth map \(F : M \to N\) between manifolds, that linear map is the
differential
\(dF_p : T_pM \to T_{F(p)}N\). The differential is a coarse instrument. For a general map,
higher-order behavior at a critical point is invisible to it. The maps \(x \mapsto x^2\) and
\(x \mapsto x^3\) on \(\mathbb{R}\) agree at the origin, where both have \(dF_0 = 0\), yet behave
quite differently. But a linear map between finite-dimensional vector spaces carries exactly one
coordinate-independent invariant, its rank, and the maps studied on this page are
precisely those for which the rank alone already dictates the local geometry.
Two linear maps of the same rank differ only by a choice of bases in domain and codomain. Nothing
else about them is intrinsic. We study the maps whose differential has the same rank at every
point, for it is these whose local behavior is rigidly modeled by their linearization.
Definition: Rank of a Smooth Map
Let \(M\) and \(N\) be smooth manifolds, with or without boundary, let \(F : M \to N\) be a
smooth map, and let \(p \in M\). The rank of \(F\) at \(p\) is the rank of
the linear map \(dF_p : T_pM \to T_{F(p)}N\). Equivalently, it is the rank of the Jacobian
matrix of \(F\) in any pair of smooth coordinate charts around \(p\) and \(F(p)\), or the
dimension of the image \(\dim\big(\operatorname{im} dF_p\big)\). If \(F\) has the same rank
\(r\) at every point, we say \(F\) has constant rank and write
\(\operatorname{rank} F = r\).
That these three descriptions agree, and that the rank does not depend on the charts chosen, is
the content of the
coordinate representation of the differential.
In charts the differential is the Jacobian, and a change of charts multiplies the
Jacobian on each side by an invertible matrix, which leaves the rank unchanged. Writing
\(m = \dim M\) and \(n = \dim N\), the rank of a linear map \(T_pM \to T_{F(p)}N\) can never
exceed \(\min\{m, n\}\). When it attains this maximum we call \(F\) full rank (at
\(p\), or everywhere). Two extreme cases of full rank organize the entire theory.
Definition: Smooth Submersion
A smooth map \(F : M \to N\) between smooth manifolds, with or without boundary, is a
smooth submersion if its differential \(dF_p\) is surjective
at every point \(p \in M\). Equivalently, \(F\) has constant rank equal to \(\dim N\).
Definition: Smooth Immersion
A smooth map \(F : M \to N\) between smooth manifolds, with or without boundary, is a
smooth immersion if its differential \(dF_p\) is injective
at every point \(p \in M\). Equivalently, \(F\) has constant rank equal to \(\dim M\).
A submersion has full rank \(n\), so it requires \(m \geq n\). An immersion has full rank \(m\),
so it requires \(m \leq n\). These are the two ways a map can have the largest rank its dimensions
allow, and they are the most important constant-rank maps. The equivalence with the constant-rank
condition, asserted in each definition, is immediate. Surjectivity of \(dF_p\) says exactly that
\(\operatorname{rank} dF_p = n\), and injectivity that \(\operatorname{rank} dF_p = m\), so either
condition pins the rank to the same value at every point. What is not immediate, and what makes
these two classes special, is that full rank is an open condition.
Proposition: Full Rank is an Open Condition
Let \(M\) and \(N\) be smooth manifolds, with or without boundary, and let \(F : M \to N\) be
smooth. If \(dF_p\) is surjective (respectively injective) at a point \(p\), then \(dF_q\) is
surjective (respectively injective) for all \(q\) in a neighborhood of \(p\).
Proof Sketch.
In any pair of smooth charts the differential is represented by the Jacobian matrix, and
surjectivity or injectivity of \(dF_p\) is exactly the statement that this matrix has full
rank at \(p\). The set of full-rank matrices is open in the space of all matrices. A matrix
has full rank precisely when some maximal minor is nonzero, and that minor is a continuous
(indeed polynomial) function of the entries, so the condition persists on a neighborhood. This
is recorded for matrices in the proposition that
full-rank matrices form an open set.
Since the Jacobian entries depend continuously on the base point, full rank at \(p\)
propagates to a neighborhood of \(p\).
An asymmetry worth noting
Full rank is open and therefore locally stable. An immersion or submersion stays one
under small perturbations and on neighborhoods. Constant rank that is not full rank enjoys no
such protection. The rank can drop on a closed set without warning. The map
\((x, y) \mapsto (x^2, 0)\) on \(\mathbb{R}^2\), for instance, has Jacobian
\(\left(\begin{smallmatrix} 2x & 0 \\\\ 0 & 0 \end{smallmatrix}\right)\), of rank \(1\)
wherever \(x \neq 0\) but rank \(0\) on the entire line \(\{x = 0\}\). This is exactly why the
two full-rank classes, immersions and submersions, are singled out. Their defining condition
is the robust one.
The basic examples
A handful of examples fix the picture and recur throughout this page.
Projections are submersions.
For smooth manifolds \(M_1, \dots, M_k\), each
projection \(\pi_i : M_1 \times \cdots \times M_k \to M_i\) is a smooth submersion. In coordinates
it discards some of the factors, so its differential discards the corresponding tangent directions
and is therefore surjective. The prototype is the coordinate projection
\(\pi : \mathbb{R}^{n+k} \to \mathbb{R}^n\) onto the first \(n\) coordinates.
Curves are immersions exactly when their velocity never vanishes.
A smooth
curve \(\gamma : J \to M\) (with \(J\) an interval, possibly with boundary) is a smooth immersion
if and only if its
velocity
\(\gamma'(t)\) is nonzero for every \(t \in J\). Here \(\dim J = 1\), so full rank means
\(\operatorname{rank} = 1\), and the single column of the Jacobian is precisely the velocity
vector. The differential is injective exactly when that column is nonzero.
The tangent bundle projection is a submersion.
When the
tangent bundle
\(TM\) carries its natural smooth structure, the projection \(\pi : TM \to M\) sending a tangent
vector to its base point is a smooth submersion. In the natural coordinates \((x^i, v^i)\) on
\(\pi^{-1}(U) \subseteq TM\) induced by a chart \((U, \varphi)\), the coordinate representation of
\(\pi\) is simply \(\widehat\pi(x, v) = x\), which is a coordinate projection and so a submersion.
An immersion that is not injective.
The map
\(X : \mathbb{R}^2 \to \mathbb{R}^3\),
\[
X(u, v) = \big((2 + \cos 2\pi u)\cos 2\pi v,\ (2 + \cos 2\pi u)\sin 2\pi v,\ \sin 2\pi u\big),
\]
is a smooth immersion whose image is the doughnut-shaped surface obtained by revolving the circle
\((y - 2)^2 + z^2 = 1\) in the \((y, z)\)-plane about the \(z\)-axis. Its differential is
injective everywhere, yet \(X\) is far from injective. The map is periodic in both arguments. This
separation between the pointwise condition (immersion) and the global one (injectivity) is the
theme that the next two sections develop in full.
Finally, constant rank is preserved under composition only in the full-rank cases. A composition
of submersions is a submersion and a composition of immersions is an immersion, since surjectivity
and injectivity of linear maps compose. A composition of two constant-rank maps that are not full
rank need not have constant rank at all, another reflection of the asymmetry noted above.
Immersions, Submersions, and Local Diffeomorphisms
The most regular maps of all are those whose differential is an isomorphism at every point, which
is full rank in both senses at once and is possible only when \(\dim M = \dim N\). Such a map
looks, near each point, exactly like a diffeomorphism, even if it fails to be one globally. This
is the manifold incarnation of the inverse function theorem, and it is the bridge between the
linear-algebraic notion of rank and the genuinely nonlinear statement that a map can be inverted.
Definition: Local Diffeomorphism
Let \(M\) and \(N\) be smooth manifolds, with or without boundary. A map \(F : M \to N\) is a
local diffeomorphism if every point \(p \in M\) has a neighborhood \(U\) such
that \(F(U)\) is open in \(N\) and \(F|_U : U \to F(U)\) is a diffeomorphism.
A diffeomorphism is a local diffeomorphism, but the converse fails. The map
\(\varepsilon : \mathbb{R} \to S^1\), \(\varepsilon(t) = (\cos 2\pi t, \sin 2\pi t)\), restricts
to a diffeomorphism on any interval of length less than \(1\), yet wraps the line infinitely many
times around the circle and is nowhere injective globally. We return to this map, and to its
higher analogue \(\varepsilon^n : \mathbb{R}^n \to T^n\), when we study smooth covering maps. The
next theorem is the engine behind every local diffeomorphism. It promotes invertibility of the
differential at a point into a local diffeomorphism near that point.
Theorem (Inverse Function Theorem for Manifolds)
Suppose \(M\) and \(N\) are smooth manifolds (without boundary), and \(F : M \to N\) is a
smooth map. If \(p \in M\) is a point such that \(dF_p\) is invertible, then there are connected
neighborhoods \(U_0\) of \(p\) and \(V_0\) of \(F(p)\) such that \(F|_{U_0} : U_0 \to V_0\) is a
diffeomorphism.
Proof Sketch.
Invertibility of \(dF_p\) forces \(\dim M = \dim N =: n\). Choose smooth charts
\((U, \varphi)\) centered at \(p\) and \((V, \psi)\) centered at \(F(p)\) with
\(F(U) \subseteq V\), and pass to the coordinate representation
\(\widehat F = \psi \circ F \circ \varphi^{-1}\), a smooth map between open subsets of
\(\mathbb{R}^n\) with \(\widehat F(0) = 0\). Since \(\varphi\) and \(\psi\) are
diffeomorphisms, the chain rule gives
\(d\widehat F_0 = d\psi_{F(p)} \circ dF_p \circ d(\varphi^{-1})_0\), a composition of
isomorphisms, hence nonsingular.
The
Euclidean inverse function theorem
then restricts \(\widehat F\) to a diffeomorphism between connected open neighborhoods of the
origin. Pulling these back through \(\varphi\) and \(\psi\) and composing yields the desired
diffeomorphism \(F|_{U_0}\).
The hypothesis that \(M\) and \(N\) have empty boundary is essential. The theorem can fail when
the domain has nonempty boundary. When only the codomain has boundary, the conclusion survives
provided \(F\) takes its values in the interior, which is automatic at any point where \(dF_p\) is
invertible.
Local diffeomorphisms inherit the convenient closure properties one would expect of a class defined
by a local condition.
Proposition: Elementary Properties of Local Diffeomorphisms
Let \(M\), \(N\), and \(P\) be smooth manifolds, with or without boundary.
(a) Every composition of local diffeomorphisms is a local diffeomorphism.
(b) Every finite product of local diffeomorphisms between smooth manifolds is a local
diffeomorphism.
(c) Every local diffeomorphism is a local homeomorphism and an
open map.
(d) The restriction of a local diffeomorphism to an open submanifold (with or without boundary)
is a local diffeomorphism.
(e) Every diffeomorphism is a local diffeomorphism.
(f) Every bijective local diffeomorphism is a diffeomorphism.
(g) A map is a local diffeomorphism if and only if, in a neighborhood of each point of its
domain, it has a coordinate representation that is a local diffeomorphism.
Proof Sketch.
Each statement follows directly from the definition together with the corresponding property
of diffeomorphisms. For (a), (b), (d), and (e), restrict to neighborhoods on which the maps
are honest diffeomorphisms and use that compositions, products, restrictions, and the maps
themselves are diffeomorphisms there. For (c), a diffeomorphism is in particular a
homeomorphism, and a map that is locally a homeomorphism onto open sets carries open sets to
open sets.
For (f), let \(F\) be a bijective local diffeomorphism. By (c) it is in particular a local
homeomorphism and an open map, and an open continuous bijection is a homeomorphism, so the
set-theoretic inverse \(F^{-1}\) is continuous. To see that \(F^{-1}\) is smooth, fix
\(q \in N\) and let \(p = F^{-1}(q)\). Choose a neighborhood \(U\) of \(p\) on which
\(F|_U : U \to F(U)\) is a diffeomorphism. Since \(F\) is an open map by (c), the set \(F(U)\)
is a neighborhood of \(q\) on which \(F^{-1}\) coincides with the smooth map \((F|_U)^{-1}\).
Thus \(F^{-1}\) is smooth on a neighborhood of each point, and since smoothness is a local
property, \(F^{-1}\) is smooth. Therefore \(F\) is a diffeomorphism. For (g), the property of
being a local diffeomorphism is itself local and is detected in charts, since charts are
diffeomorphisms.
The payoff is the precise relationship between local diffeomorphisms and the two full-rank classes.
A local diffeomorphism is exactly a map that is simultaneously as injective and as surjective on
tangent spaces as it can be.
Proposition: Local Diffeomorphisms vs. Immersions and Submersions
Suppose \(M\) and \(N\) are smooth manifolds (without boundary) and \(F : M \to N\) is a map.
(a) \(F\) is a local diffeomorphism if and only if it is both a smooth immersion and a smooth
submersion.
(b) If \(\dim M = \dim N\) and \(F\) is either a smooth immersion or a smooth submersion, then
it is a local diffeomorphism.
Proof Sketch.
For (a), suppose \(F\) is a local diffeomorphism. Near each point it restricts to a
diffeomorphism between open submanifolds, whose differential is an isomorphism by the
properties of the differential.
Since the
tangent space to an open submanifold
is canonically identified with the ambient one, \(dF_p\) is an isomorphism for every \(p\), in
particular both injective and surjective, so \(F\) is at once an immersion and a submersion.
Conversely, if \(F\) is both, then \(dF_p\) is injective and surjective, hence an isomorphism
at every point, and the
inverse function theorem for manifolds
makes \(F\) a local diffeomorphism. For (b), when \(\dim M = \dim N\) a linear map between the
equidimensional tangent spaces is injective if and only if it is surjective if and only if it
is an isomorphism. Either hypothesis therefore makes \(dF_p\) an isomorphism, and (a) applies.
Thus, in equal dimensions, "immersion," "submersion," and "local diffeomorphism" all coincide. The
distinctions among them matter only when \(\dim M \neq \dim N\).
Two further remarks frame what follows. First, the theorem of this section is purely local. It
says nothing about whether \(F\) is globally injective or whether its image is well behaved. The
map \(\varepsilon\) above is a local diffeomorphism that is neither injective nor a homeomorphism
onto its image, and recovering global statements from local ones is precisely the work of the next
two sections. Second, both results here describe the full-rank case \(r = \min\{m, n\}\). The
genuinely new phenomenon, that any constant rank, not only the full one, forces a rigid
local normal form, is the subject of the rank theorem, to which we now turn.
The Rank Theorem
We come to the central fact about constant-rank maps, and the technical heart of this page. In
linear algebra, a linear map of rank \(r\) can be put into the canonical form
\((x^1, \dots, x^m) \mapsto (x^1, \dots, x^r, 0, \dots, 0)\) by suitable choices of basis in
domain and codomain. Its rank is its only invariant, and once the rank is fixed the map is
determined up to change of basis.
The rank theorem is the nonlinear counterpart. A smooth map of constant rank \(r\) can be
brought into exactly this same form, not by linear changes of basis, but by smooth changes of
coordinates near each point. It is a direct consequence of the inverse function theorem,
and everything later on this page rests on it, from embeddings and submersions to the global rank
theorem.
Theorem (Rank Theorem)
Suppose \(M\) and \(N\) are smooth manifolds (without boundary) of dimensions \(m\) and \(n\),
and \(F : M \to N\) is a smooth map with constant rank \(r\). For each \(p \in M\) there exist
smooth charts \((U, \varphi)\) centered at \(p\) and \((V, \psi)\) centered at \(F(p)\), with
\(F(U) \subseteq V\), in which \(F\) has the coordinate representation
\[
\widehat F\big(x^1, \dots, x^r, x^{r+1}, \dots, x^m\big) = \big(x^1, \dots, x^r, 0, \dots, 0\big).
\]
In particular, a smooth submersion (\(r = n\)) has a local representation
\(\widehat F(x^1, \dots, x^n, x^{n+1}, \dots, x^m) = (x^1, \dots, x^n)\), and a smooth
immersion (\(r = m\)) has a local representation
\(\widehat F(x^1, \dots, x^m) = (x^1, \dots, x^m, 0, \dots, 0)\).
Proof Sketch.
The statement is local, so working in charts we may assume \(F\) maps an open subset of
\(\mathbb{R}^m\) to an open subset of \(\mathbb{R}^n\), with \(p\) and \(F(p)\) at the
origins. Because \(DF(p)\) has rank \(r\), some \(r \times r\) minor is nonsingular. After
permuting coordinates we may take it to be the upper-left block. Split the coordinates as
\((x, y)\) with \(x \in \mathbb{R}^r\) and write \(F(x, y) = (Q(x, y), R(x, y))\) with \(Q\)
valued in \(\mathbb{R}^r\), so that \((\partial Q^i / \partial x^j)\) is nonsingular at the
origin.
Straightening the domain.
Define \(\varphi(x, y) = (Q(x, y), y)\). Its
Jacobian at the origin is block triangular with the nonsingular block
\((\partial Q^i / \partial x^j)\) and an identity block on the diagonal, hence nonsingular, so
the
inverse function theorem
makes \(\varphi\) a diffeomorphism on a connected neighborhood of the origin. This is where
the diffeomorphism property is earned, not assumed. Shrink that neighborhood to an
open cube. The cube will matter shortly. Writing the inverse as
\(\varphi^{-1}(x, y) = (A(x, y), B(x, y))\) and comparing the \(y\)-components of
\(\varphi \circ \varphi^{-1} = \mathrm{id}\) forces \(B(x, y) = y\), so
\(\varphi^{-1}(x, y) = (A(x, y), y)\) and \(Q(A(x, y), y) = x\).
Reading off the form.
Composing gives
\(F \circ \varphi^{-1}(x, y) = (x, \widetilde R(x, y))\), where
\(\widetilde R(x, y) = R(A(x, y), y)\). Since composing with a diffeomorphism preserves rank,
this map still has rank \(r\) throughout the cube. Its Jacobian has an identity block in its
first \(r\) columns, which are independent. For the total rank to remain \(r\) the block
\(\partial \widetilde R^i / \partial y^j\) must vanish identically. On a cube every
segment parallel to a \(y\)-axis stays inside the domain, so vanishing \(y\)-derivatives mean
that \(\widetilde R\) does not depend on \(y\) at all. Setting \(S(x) = \widetilde R(x, 0)\)
gives \(F \circ \varphi^{-1}(x, y) = (x, S(x))\).
Straightening the codomain.
Finally define \(\psi(v, w) = (v, w - S(v))\)
on a suitable neighborhood of the origin in \(\mathbb{R}^n\). It is a diffeomorphism, with
explicit smooth inverse \(\psi^{-1}(s, t) = (s, t + S(s))\), so \((\,\cdot\,, \psi)\) is a
smooth chart. Composing gives
\(\psi \circ F \circ \varphi^{-1}(x, y) = \psi(x, S(x)) = (x, S(x) - S(x)) = (x, 0)\), which
is the asserted normal form.
What the cube is for
The construction has four moving parts, and the constant-rank hypothesis is what makes the
third one work. Surjectivity or injectivity of a single differential would straighten the map
at one point. The assumption that the rank stays equal to \(r\) on a whole neighborhood is
what forces \(\partial \widetilde R / \partial y \equiv 0\). The geometric convexity of a cube
is what upgrades "the \(y\)-derivative vanishes" into "\(\widetilde R\) is genuinely
independent of \(y\)."
Each piece earns its place: the diffeomorphism \(\varphi\) comes from the inverse function
theorem, the identity \(B(x, y) = y\) from comparing components, the \(y\)-independence of
\(\widetilde R\) from rank invariance plus the cube, and the diffeomorphism \(\psi\) from an
inverse written down by hand.
The rank theorem admits a coordinate-free reformulation that says, in a single phrase, what kind
of maps the constant-rank maps are. They are precisely the maps whose local behavior is
indistinguishable from that of their differential.
Corollary: Constant Rank Means Locally Linear
Let \(M\) and \(N\) be smooth manifolds (without boundary), let \(F : M \to N\) be a smooth
map, and suppose \(M\) is connected. The following are equivalent:
(a) For each \(p \in M\) there exist smooth charts around \(p\) and \(F(p)\) in which the
coordinate representation of \(F\) is linear.
(b) \(F\) has constant rank.
Proof Sketch.
If \(F\) is linear in suitable charts near each point, then near each point its rank equals
the rank of that linear map, so the rank is locally constant. On a connected manifold a
locally constant integer-valued function is globally constant, giving (b). Conversely, if
\(F\) has constant rank, the normal form of the rank theorem is itself a linear coordinate
representation, giving (a).
Global and Boundary Versions
The rank theorem is purely local, but it has a powerful global consequence. For a constant-rank
map, a single global property decides the pointwise one. Whether the map is surjective or
injective determines whether its differential is everywhere surjective or everywhere injective.
This is the most striking payoff of the constant-rank hypothesis. Global set-theoretic behavior
dictates local differential behavior.
Theorem (Global Rank Theorem)
Let \(M\) and \(N\) be smooth manifolds (without boundary), and suppose \(F : M \to N\) is a
smooth map of constant rank.
(a) If \(F\) is surjective, then it is a smooth submersion.
(b) If \(F\) is injective, then it is a smooth immersion.
(c) If \(F\) is bijective, then it is a diffeomorphism.
Proof Sketch.
Write \(m = \dim M\), \(n = \dim N\), and let \(r\) be the constant rank.
(a)
We prove the contrapositive. If \(F\) is not a submersion, it cannot
be surjective. Suppose \(r \lt n\). The rank theorem provides, around each \(p\), a chart in
which \(F\) looks like \((x^1, \dots, x^r, 0, \dots, 0)\). Shrink the domain chart to a
regular coordinate ball
\(U\) with \(F(\overline U) \subseteq V\), which is available because such balls
form a basis.
The image \(F(\overline U)\) is then a compact subset of the slice
\(\{y^{r+1} = \cdots = y^n = 0\}\). That slice contains no open subset of \(N\) (it has
positive codimension), and a compact set is closed, so \(F(\overline U)\) is closed with empty
interior, that is, nowhere dense. Because a manifold is
second-countable,
every open cover admits a countable subcover,
so countably many such balls \(\{U_i\}\) cover \(M\), and then
\(F(M) = \bigcup_i F(\overline{U_i})\) is a countable union of nowhere dense sets. By the
Baire category theorem,
applicable because \(N\) is Hausdorff and
locally compact,
this union has empty interior. In particular it is not all of \(N\), so \(F\) is not
surjective.
(b)
Again by contrapositive. If \(r \lt m\), the rank theorem gives a
chart in which \(F(x^1, \dots, x^m) = (x^1, \dots, x^r, 0, \dots, 0)\), a representation that
ignores the last \(m - r \geq 1\) coordinates. Then for any sufficiently small
\(\varepsilon\), \(F(0, \dots, 0, \varepsilon) = F(0, \dots, 0, 0)\), two distinct points with
the same image, so \(F\) is not injective.
(c)
A bijective constant-rank map is surjective and injective, hence by (a) and
(b) both a submersion and an immersion. By the
characterization of local diffeomorphisms
it is a local diffeomorphism, and a
bijective local diffeomorphism is a diffeomorphism.
Two independent reasons a constant-rank map can fail to be onto or one-to-one
The two halves of the proof are different in character. In (a), the key is not merely that
each \(F(\overline U)\) sits in a lower-dimensional slice, but that it is
nowhere dense, being closed by compactness and of empty interior by positive
codimension, and that the whole image \(F(M)\) is a countable union of such
pieces, the countability coming from the existence of a countable subcover. Only with both
facts in hand does Baire's theorem apply to conclude that \(F(M)\) misses an open set.
In (b), the failure of injectivity is local and explicit. The normal form simply forgets
coordinates, and forgotten coordinates produce collisions. The surjective case is a theorem
about the size of the image. The injective case is a theorem about the form of the map.
The boundary case
For manifolds with boundary the rank theorem is needed in exactly one situation: a smooth immersion
whose domain is a manifold with boundary. Away from the boundary nothing new arises, since the
interior of a manifold with boundary is an ordinary smooth manifold and the rank theorem already
applies there. The substitute at boundary points is the following.
Theorem (Local Immersion Theorem for Manifolds with Boundary)
Suppose \(M\) is a smooth \(m\)-manifold with boundary, \(N\) is a smooth \(n\)-manifold, and
\(F : M \to N\) is a smooth immersion. For any \(p \in \partial M\), there exist a smooth
boundary chart \((U, \varphi)\) for \(M\) centered at \(p\) and a smooth coordinate chart
\((V, \psi)\) for \(N\) centered at \(F(p)\), with \(F(U) \subseteq V\), in which \(F\) has the
coordinate representation
\[
\widehat F\big(x^1, \dots, x^m\big) = \big(x^1, \dots, x^m, 0, \dots, 0\big).
\]
Proof Sketch.
Working in initial charts we may take \(M\) and \(N\) to be open subsets of \(\mathbb{H}^m\)
and \(\mathbb{R}^n\), with \(p = 0\) and \(F(p) = 0\). By the
definition of smoothness on a half-space,
\(F\) extends to a smooth map \(\widetilde F\) on an open subset \(W \subseteq \mathbb{R}^m\)
containing \(0\). Since \(d\widetilde F_0 = dF_0\) is injective, the fact that
full rank is an open condition
lets us shrink \(W\) so that \(\widetilde F\) is a smooth immersion there, and the ordinary
rank theorem
(immersion case) furnishes charts \((U_0, \varphi_0)\) and \((V_0, \psi_0)\) in which
\(\widetilde F\) takes the form \(x \mapsto (x, 0)\).
The one defect is that \(\varphi_0\), built for the extended map on an open subset of
\(\mathbb{R}^m\), need not restrict to a boundary chart for \(M\). This is repaired by a
single adjustment. Since \(\varphi_0\) is a diffeomorphism onto an open set
\(\widehat U_0 \subseteq \mathbb{R}^m\), the map
\(\varphi_0^{-1} \times \operatorname{Id}_{\mathbb{R}^{n-m}}\) is a diffeomorphism of
\(\widehat U_0 \times \mathbb{R}^{n-m}\) onto \(U_0 \times \mathbb{R}^{n-m}\). After shrinking
\(V_0\) so that \(\psi_0(V_0)\) lies in \(\widehat U_0 \times \mathbb{R}^{n-m}\), composing
the codomain chart with it yields a corrected chart
\(\psi = \big(\varphi_0^{-1} \times \operatorname{Id}_{\mathbb{R}^{n-m}}\big) \circ \psi_0\).
A direct computation using the immersion normal form then gives \(\psi \circ F(x) = (x, 0)\),
so the original \(M\)-coordinates (restricted to a small enough neighborhood) together with
\((V, \psi)\) realize the asserted form.
Two limitations of the boundary version reflect genuine obstructions rather than gaps in the
argument. First, the theorem concerns immersions out of a manifold with boundary only. A
constant-rank version for general ranks can be proved, but is more delicate, because an extension
of a constant-rank map across the boundary need not retain constant rank. We follow the standard
treatment in not pursuing it.
Second, the situation for a map whose codomain has boundary is markedly worse. The image
can meet the boundary in unpredictable ways, and no canonical form exists without strong
additional hypotheses. The clean theory developed here is, by design, a theory of boundaryless
targets.
With the rank theorem and its global and boundary consequences in hand, we possess the full
local-to-global machinery for constant-rank maps. The two full-rank classes now branch into the
central constructions of the smooth category. Immersions, refined by topological hypotheses,
become the embeddings that carve out submanifolds. Submersions, paired with the section theorem,
become the smooth analogues of quotient maps. Developing these two threads is the work that
follows.