The Smooth Manifold Chart Lemma
All of the smooth structures built so far on this site have followed the same two-stage recipe. First, a set is endowed with
a topology, and the three conditions of a
topological manifold
— Hausdorff, second-countable, locally Euclidean — are verified directly. Then, with the topology in hand, an explicit atlas
is exhibited whose transition maps are checked to be smooth, and the
smooth-structure-from-atlas proposition
converts that atlas into a smooth structure. The hemisphere atlas of \(\mathbb{S}^n\), the affine atlas of \(\mathbb{RP}^n\),
and the product atlas of \(\mathbb{T}^n\) were all produced this way on the
previous page.
The two-stage recipe is conceptually clean, but it presupposes that a natural topology is already available on the set.
For some constructions — most prominently the Grassmann manifolds taken up in the next section — the most natural starting
point is not a topological space at all, but a bare set together with a candidate collection of charts. One would like a
shortcut: a single criterion on the candidate charts that simultaneously builds a topology and the smooth structure on the
set. The lemma below provides exactly such a shortcut.
Statement of the Lemma
The hypotheses ask for five properties of a candidate chart collection. The first three are local — they govern how the
charts overlap and what their transition maps look like — while the last two are global, controlling the size and
separation properties of the resulting space.
Lemma: Smooth Manifold Chart Lemma
Let \(M\) be a set, and suppose we are given a collection \(\{U_\alpha\}\) of subsets of \(M\) together with maps
\(\varphi_\alpha : U_\alpha \to \mathbb{R}^n\), such that the following properties are satisfied:
-
For each \(\alpha\), \(\varphi_\alpha\) is a bijection between \(U_\alpha\) and an open subset
\(\varphi_\alpha(U_\alpha) \subseteq \mathbb{R}^n\).
-
For each \(\alpha\) and \(\beta\), the sets \(\varphi_\alpha(U_\alpha \cap U_\beta)\) and
\(\varphi_\beta(U_\alpha \cap U_\beta)\) are open in \(\mathbb{R}^n\).
-
Whenever \(U_\alpha \cap U_\beta \ne \emptyset\), the map
\(\varphi_\beta \circ \varphi_\alpha^{-1} : \varphi_\alpha(U_\alpha \cap U_\beta) \to \varphi_\beta(U_\alpha \cap U_\beta)\)
is smooth.
-
Countably many of the sets \(U_\alpha\) cover \(M\).
-
Whenever \(p, q\) are distinct points in \(M\), either there exists some \(U_\alpha\) containing both \(p\) and
\(q\), or there exist disjoint sets \(U_\alpha, U_\beta\) with \(p \in U_\alpha\) and \(q \in U_\beta\).
Then \(M\) has a unique smooth manifold structure such that each \((U_\alpha, \varphi_\alpha)\) is a
smooth chart.
A few words on the conditions before the proof. Conditions (i) and (ii) are minor regularity demands — bijectivity is what
a chart should provide, and openness of \(\varphi_\alpha(U_\alpha)\) and of the chart images of overlaps are necessary if
the result is to look like Euclidean space locally. Condition (iii) is the substantive smoothness condition, identical to
the
smooth compatibility
requirement of an ordinary smooth atlas, except that the transition maps now live in \(\mathbb{R}^n\) by virtue of (i) and
(ii), without any prior topology on \(M\). Conditions (iv) and (v) are precisely what is needed to deduce
second-countability and Hausdorffness of the topology that will be constructed: (iv) plays the role of a countable basis
condition, and (v) is a chart-level rephrasing of the Hausdorff axiom.
Proof of the Lemma
The proof has three movements: build the topology, verify that it makes \(M\) a topological manifold, and verify that the
candidate charts form a smooth atlas.
Proof:
Topology.
Declare the collection
\[
\mathcal{B} = \{\varphi_\alpha^{-1}(V) : \alpha \text{ any index},\ V \text{ open in } \mathbb{R}^n\}
\]
to be a basis of a topology on \(M\). To verify that this is in fact a basis, two conditions must be checked: that the
sets in \(\mathcal{B}\) cover \(M\), and that the intersection of any two sets in \(\mathcal{B}\) is a union of sets
in \(\mathcal{B}\). The cover condition is immediate: by (iv), the \(U_\alpha\) cover \(M\), and each \(U_\alpha\) is
itself a basis set (taking \(V = \varphi_\alpha(U_\alpha)\)).
For the intersection condition, fix two basis sets \(\varphi_\alpha^{-1}(V)\) and \(\varphi_\beta^{-1}(W)\) with
\(V, W\) open in \(\mathbb{R}^n\). The key identity is
\[
\varphi_\alpha^{-1}(V) \cap \varphi_\beta^{-1}(W)
= \varphi_\alpha^{-1}\bigl(V \cap (\varphi_\beta \circ \varphi_\alpha^{-1})^{-1}(W)\bigr),
\]
which holds for the following reason. A point \(p\) lies in the left-hand side exactly when \(p \in U_\alpha\) with
\(\varphi_\alpha(p) \in V\) and simultaneously \(p \in U_\beta\) with \(\varphi_\beta(p) \in W\); the second condition
rewrites as \(\varphi_\beta \circ \varphi_\alpha^{-1}(\varphi_\alpha(p)) \in W\), i.e., \(\varphi_\alpha(p)\) lies in
the preimage of \(W\) under the transition. Both conditions together require \(\varphi_\alpha(p)\) to lie in
\(V \cap (\varphi_\beta \circ \varphi_\alpha^{-1})^{-1}(W)\), which is the right-hand side.
It remains to show that the right-hand side is itself a basis set — i.e., that the set
\(V \cap (\varphi_\beta \circ \varphi_\alpha^{-1})^{-1}(W)\) is an open subset of \(\mathbb{R}^n\). By (iii), the
transition \(\varphi_\beta \circ \varphi_\alpha^{-1}\) is smooth on
\(\varphi_\alpha(U_\alpha \cap U_\beta)\); smooth maps are continuous, so the preimage
\((\varphi_\beta \circ \varphi_\alpha^{-1})^{-1}(W)\) is open in \(\varphi_\alpha(U_\alpha \cap U_\beta)\). By (ii),
\(\varphi_\alpha(U_\alpha \cap U_\beta)\) is itself open in \(\mathbb{R}^n\), so the preimage is open in
\(\mathbb{R}^n\) as well. Intersecting with the open set \(V\) preserves openness. The intersection condition is
verified, and \(\mathcal{B}\) is a basis. The resulting topology on \(M\) is the one in which the basis sets, and all
their unions, are declared open.
Each \(\varphi_\alpha\) is a homeomorphism onto its image.
By (i), \(\varphi_\alpha\) is a bijection from \(U_\alpha\) to \(\varphi_\alpha(U_\alpha)\). The basis sets contained
in \(U_\alpha\) are exactly the sets of the form \(\varphi_\alpha^{-1}(V)\) with \(V \subseteq \varphi_\alpha(U_\alpha)\)
open, and these are by construction the open subsets of \(U_\alpha\) in the subspace topology. The map
\(\varphi_\alpha\) sends each such \(\varphi_\alpha^{-1}(V)\) to \(V\), and is therefore an open map; its inverse sends
each open \(V \subseteq \varphi_\alpha(U_\alpha)\) to the basis set \(\varphi_\alpha^{-1}(V)\), and is therefore
continuous. Continuity of \(\varphi_\alpha\) itself follows by the same argument with the roles reversed.
\(M\) is locally Euclidean of dimension \(n\).
Every point of \(M\) lies in some \(U_\alpha\) by (iv), and on each \(U_\alpha\) the map \(\varphi_\alpha\) gives a
homeomorphism to the open subset \(\varphi_\alpha(U_\alpha) \subseteq \mathbb{R}^n\). So every point has a
neighborhood homeomorphic to an open subset of \(\mathbb{R}^n\).
\(M\) is Hausdorff.
Fix distinct points \(p, q \in M\). By (v), either both lie in some common \(U_\alpha\), or they lie in disjoint
\(U_\alpha\) and \(U_\beta\). In the second case, \(U_\alpha\) and \(U_\beta\) are open in \(M\) (each is a basis set)
and disjoint, so they separate \(p\) from \(q\). In the first case, both \(p\) and \(q\) lie in \(U_\alpha\) with
\(\varphi_\alpha(p) \ne \varphi_\alpha(q)\) (because \(\varphi_\alpha\) is a bijection); the Hausdorff property of
\(\mathbb{R}^n\) provides disjoint open neighborhoods of \(\varphi_\alpha(p)\) and \(\varphi_\alpha(q)\), and their
preimages under \(\varphi_\alpha\) are disjoint basis-set neighborhoods of \(p\) and \(q\) in \(M\).
\(M\) is second-countable.
By (iv), a countable subcollection of the \(U_\alpha\) covers \(M\); call this subcollection
\(\{U_{\alpha_k}\}_{k \in \mathbb{N}}\). Each \(U_{\alpha_k}\) is homeomorphic, via \(\varphi_{\alpha_k}\), to an open
subset of \(\mathbb{R}^n\), and is therefore second-countable: writing \(\mathcal{R}\) for the countable basis of
\(\mathbb{R}^n\) consisting of open balls with rational centers and rational radii, the collection
\[
\mathcal{B}_k = \{\varphi_{\alpha_k}^{-1}(R) : R \in \mathcal{R},\ R \subseteq \varphi_{\alpha_k}(U_{\alpha_k})\}
\]
is a countable basis for the subspace topology on \(U_{\alpha_k}\). We claim that the union
\(\mathcal{C} = \bigcup_{k} \mathcal{B}_k\), a countable collection of open subsets of \(M\), is a basis for the
topology of \(M\).
To verify the claim, let \(O \subseteq M\) be open and let \(p \in O\). Since the \(U_{\alpha_k}\) cover \(M\),
there is an index \(k\) with \(p \in U_{\alpha_k}\). The set \(O \cap U_{\alpha_k}\) is open in \(M\) and contained
in \(U_{\alpha_k}\), hence open in the subspace topology on \(U_{\alpha_k}\); because \(\mathcal{B}_k\) is a basis for
that subspace topology, there is a member \(B \in \mathcal{B}_k\) with \(p \in B \subseteq O \cap U_{\alpha_k} \subseteq O\).
Thus every point of every open set \(O\) lies in a member of \(\mathcal{C}\) contained in \(O\), which is exactly the
condition for \(\mathcal{C}\) to be a basis for the topology of \(M\). Since \(\mathcal{C}\) is countable, \(M\) is
second-countable.
\(\{(U_\alpha, \varphi_\alpha)\}\) is a smooth atlas.
The previous steps make each \((U_\alpha, \varphi_\alpha)\) a chart on \(M\) in the topological sense. By (iii), the
transition maps \(\varphi_\beta \circ \varphi_\alpha^{-1}\) between any two charts with overlapping domains are
smooth; symmetrically (interchanging \(\alpha\) and \(\beta\)), the reverse transitions are smooth, so each transition
is a diffeomorphism. The atlas is therefore a smooth atlas. By the
smooth-structure-from-atlas proposition, it determines a unique
smooth structure
on \(M\) containing it, in which each \((U_\alpha, \varphi_\alpha)\) is a smooth chart.
Uniqueness.
The topology constructed above is the unique topology on \(M\) in which each \(\varphi_\alpha\) is a homeomorphism
onto an open subset of \(\mathbb{R}^n\): any such topology must contain each \(\varphi_\alpha^{-1}(V)\) as an open
set, hence the basis \(\mathcal{B}\); conversely, any topology generated by \(\mathcal{B}\) makes each
\(\varphi_\alpha\) a homeomorphism onto its image, by the second movement above. With the topology fixed, the smooth
structure containing \(\{(U_\alpha, \varphi_\alpha)\}\) is also unique by the smooth-structure-from-atlas proposition.
\(\blacksquare\)
The chart lemma is the fourth and most flexible of the construction techniques developed so far. The earlier three —
graphs,
quotients, and
products
— each require the input data to come pre-equipped with a topology compatible with the construction. The chart lemma
requires no such input: the charts themselves generate the topology. The next section uses this freedom decisively, in a
setting where the natural object is a set of subspaces and there is no obvious topology to start from.
Grassmannians
The chart lemma was developed in the abstract, and it is time to put it to decisive use. The set on which it operates
in this section is one for which no natural topology is available at the outset: the set of all \(k\)-dimensional linear
subspaces of a fixed finite-dimensional vector space. This set carries no topology from any single ambient Euclidean space
— its points are not vectors but subspaces — and any topology has to be supplied by the construction itself. The chart
lemma is the tool that supplies it, and the resulting manifold is the central object of subspace geometry.
Real projective space is the simplest example of this family: a point of \(\mathbb{RP}^n\)
is a line through the origin in \(\mathbb{R}^{n+1}\), and a line is a one-dimensional linear subspace. The
smooth projective space
constructed on the previous page is, in this language, the Grassmannian \(G_1(\mathbb{R}^{n+1})\). The general
construction below allows the subspaces to have any fixed dimension \(k\), giving a manifold that contains \(\mathbb{RP}^n\)
as its \(k = 1\) special case. Two further motivations frame the construction. Historically, the spaces of subspaces of a
fixed vector space were studied by Plücker and Grassmann in the nineteenth century as the natural setting for "subspace
coordinates," extending point and line coordinates to higher-dimensional flats. In modern applications, these same spaces
are the geometric setting in which principal-component analysis, attention-head subspaces in transformer architectures,
and the tangent-space picture of the manifold hypothesis all naturally live — a connection developed at the end of this
section.
The Grassmannian as a Set
Fix an \(n\)-dimensional real vector space \(V\) and an integer \(0 \le k \le n\).
Definition: Grassmann Manifold
The Grassmann manifold, or simply the Grassmannian, is the set
\[
G_k(V) = \{ S \subseteq V : S \text{ is a } k\text{-dimensional linear subspace of } V \}.
\]
In the case \(V = \mathbb{R}^n\), one writes \(G_k(\mathbb{R}^n)\), or sometimes \(G_{k,n}\) or \(G(k, n)\).
The use of the word "manifold" in the definition is anticipatory: at this point \(G_k(V)\) is only a set, and the
smooth-manifold structure is the content of the rest of this section. The case \(k = 1\) recovers projective space,
\(G_1(\mathbb{R}^{n+1}) = \mathbb{RP}^n\) — the set of lines through the origin in \(\mathbb{R}^{n+1}\). The case
\(k = n - 1\) of hyperplanes will also turn out to be isomorphic to a projective space (the projective space of the dual),
so the genuinely new examples occur in the range \(2 \le k \le n - 2\).
Before constructing charts, a rough dimension count predicts the answer. A \(k\)-dimensional subspace of \(V\) can be
specified by listing \(k\) linearly independent vectors in \(V\) — a choice of \(nk\) real parameters. But two such lists
determine the same subspace whenever they differ by an invertible change of basis of the subspace itself, a freedom of
\(k^2\) parameters (the dimension of \(GL(k, \mathbb{R})\)). Subtracting gives
\[
nk - k^2 = k(n - k),
\]
and \(k(n - k)\) is what the chart construction below will confirm.
Charts from Complementary Subspaces
The charts on \(G_k(V)\) are parametrized by choices of complementary subspaces of \(V\), and each chart identifies an
open subset of \(G_k(V)\) with a vector space of linear maps.
Let \(P, Q \subseteq V\) be a pair of complementary subspaces, by which we mean linear subspaces with
\(\dim P = k\), \(\dim Q = n - k\), and \(V = P \oplus Q\) (every vector of \(V\) decomposes uniquely as a sum of a vector
in \(P\) and a vector in \(Q\)). To each linear map \(X : P \to Q\) we associate its graph,
\[
\Gamma(X) = \{ v + Xv : v \in P \} \subseteq V,
\]
a \(k\)-dimensional subspace of \(V\) (the map \(v \mapsto v + Xv\) is an injective linear map from \(P\) into \(V\), so
its image has dimension equal to \(\dim P = k\)). The graph \(\Gamma(X)\) has the further property that
\(\Gamma(X) \cap Q = \{0\}\): if \(v + Xv \in Q\), then projecting along the decomposition \(V = P \oplus Q\) gives
\(v = 0\), hence \(v + Xv = 0\).
The construction reverses. Every \(k\)-dimensional subspace \(S \subseteq V\) with \(S \cap Q = \{0\}\) arises as the
graph of a unique linear map \(X : P \to Q\). To see this, let \(\pi_P : V \to P\) and \(\pi_Q : V \to Q\) be the
projections associated to the decomposition \(V = P \oplus Q\) — the linear maps sending each \(v \in V\) to its \(P\)-
and \(Q\)-components. The restriction \(\pi_P|_S : S \to P\) is then a linear isomorphism: its kernel is \(S \cap Q\),
which is \(\{0\}\) by hypothesis, so \(\pi_P|_S\) is injective; and a comparison of dimensions (\(\dim S = k = \dim P\))
shows it is also surjective. The composite
\[
X = (\pi_Q|_S) \circ (\pi_P|_S)^{-1} : P \to Q
\]
is then a well-defined linear map. To confirm that \(\Gamma(X) = S\), pick any \(s \in S\) and write \(s = v + w\) with
\(v = \pi_P(s) \in P\) and \(w = \pi_Q(s) \in Q\), the unique decomposition guaranteed by \(V = P \oplus Q\). Applying
the definition of \(X\) to this \(v\),
\[
Xv = (\pi_Q|_S) \circ (\pi_P|_S)^{-1}(v) = \pi_Q\bigl((\pi_P|_S)^{-1}(v)\bigr) = \pi_Q(s) = w,
\]
where the second-to-last equality uses that \((\pi_P|_S)^{-1}(v) = s\) by definition of the inverse. Hence
\(s = v + w = v + Xv \in \Gamma(X)\), showing \(S \subseteq \Gamma(X)\); a dimension comparison (\(\dim S = k = \dim \Gamma(X)\))
then gives equality.
Write \(L(P; Q)\) for the vector space of linear maps from \(P\) to \(Q\) — a real vector space of dimension
\(\dim P \cdot \dim Q = k(n - k)\), since after fixing bases its elements are represented by \((n - k) \times k\) real
matrices. Define the chart domain
\[
U_Q = \{ S \in G_k(V) : S \cap Q = \{0\} \}
\]
to be the set of \(k\)-dimensional subspaces of \(V\) whose intersection with \(Q\) is trivial. The two constructions
above — graph and recovery — together establish a bijection
\[
\Gamma : L(P; Q) \to U_Q, \qquad \varphi_Q = \Gamma^{-1} : U_Q \to L(P; Q).
\]
Note that \(U_Q\) depends only on \(Q\) (the subspace from which we are demanding triviality of intersection), not on the
particular choice of complement \(P\); a different choice of \(k\)-dimensional complement \(P\) gives a different
bijection between \(L(P; Q)\) and the same \(U_Q\). The map \(\varphi_Q\) does depend on \(P\), and we will keep that
dependence implicit in the notation when there is no ambiguity.
Fixing bases for \(P\) and \(Q\) identifies \(L(P; Q)\) with the matrix space \(M((n - k) \times k, \mathbb{R})\), and a
further identification with \(\mathbb{R}^{k(n - k)}\) places the chart image in Euclidean space of the expected dimension.
With this identification \(\varphi_Q : U_Q \to \mathbb{R}^{k(n - k)}\) is the candidate chart that the chart lemma will
eventually promote to a smooth chart on \(G_k(V)\).
Verifying the Hypotheses of the Chart Lemma
To apply the smooth manifold chart lemma to the candidate atlas \(\{(U_Q, \varphi_Q)\}\) — indexed over all pairs of
complementary subspaces \((P, Q)\) — we must verify each of the five hypotheses. We treat them in order.
Hypothesis (i): each chart is a bijection onto an open subset of \(\mathbb{R}^{k(n - k)}\).
By construction, \(\varphi_Q : U_Q \to L(P; Q)\) is a bijection, and the image is all of \(L(P; Q)\) — every linear map
\(X \in L(P; Q)\) has \(\Gamma(X) \in U_Q\). The identification \(L(P; Q) \cong \mathbb{R}^{k(n - k)}\) places this image
inside \(\mathbb{R}^{k(n - k)}\) as the whole space, which is trivially open. Hypothesis (i) holds.
Hypothesis (ii): chart-image overlaps are open.
Fix two charts \((U_Q, \varphi_Q)\) and \((U_{Q'}, \varphi_{Q'})\) corresponding to two choices of complementary pair,
\((P, Q)\) and \((P', Q')\). The image \(\varphi_Q(U_Q \cap U_{Q'})\) is the set of linear maps \(X \in L(P; Q)\) whose
graph \(\Gamma(X)\) lies in \(U_{Q'}\) — equivalently, whose graph intersects \(Q'\) trivially.
To recognize this set as open in \(L(P; Q)\), introduce the linear map
\[
I_X : P \to V, \qquad I_X(v) = v + Xv,
\]
a bijection from \(P\) onto the graph \(\Gamma(X)\). Composing with the projection \(\pi_{P'} : V \to P'\) gives the map
\(\pi_{P'} \circ I_X : P \to P'\), and the kernel of \(\pi_{P'}\) is precisely \(Q'\). It follows that the graph
\(\Gamma(X)\) intersects \(Q'\) trivially if and only if the linear map \(\pi_{P'} \circ I_X : P \to P'\) has trivial
kernel — equivalently, by the
rank-nullity theorem
applied to a map between spaces of the same dimension, if and only if it has
full rank \(k\).
Hypothesis (iii): transition maps are smooth.
This is the technical heart of the construction. With \((P, Q)\) and \((P', Q')\) as above, fix
\(X \in \varphi_Q(U_Q \cap U_{Q'})\) and let \(S = \Gamma(X) \in U_Q \cap U_{Q'}\) be the corresponding subspace. Write
\(X' = \varphi_{Q'}(S) \in L(P'; Q')\) for the chart value of \(S\) in the second chart; the goal is to express \(X'\) in
terms of \(X\) and verify that the resulting expression is smooth in the matrix entries of \(X\).
By the recovery formula applied to the chart \(\varphi_{Q'}\) (with \(P\) replaced by \(P'\) and \(Q\) by \(Q'\)),
\[
X' = (\pi_{Q'}|_S) \circ (\pi_{P'}|_S)^{-1} : P' \to Q'.
\]
To make this concrete in terms of \(X\), use that the map \(I_X : P \to V\) of hypothesis (ii) restricts to a linear
isomorphism \(I_X : P \to S\) (it is injective with image \(\Gamma(X) = S\)). Inserting this isomorphism and its inverse
rewrites \(X'\) as a composite that factors through \(P\):
\[
X' = (\pi_{Q'}|_S) \circ I_X \circ I_X^{-1} \circ (\pi_{P'}|_S)^{-1}
= (\pi_{Q'} \circ I_X) \circ (\pi_{P'} \circ I_X)^{-1},
\]
where the right-hand side is read as \(P' \to P \to Q'\): the factor \((\pi_{P'} \circ I_X)^{-1} : P' \to P\) followed by
\(\pi_{Q'} \circ I_X : P \to Q'\).
The two compositions \(\pi_{P'} \circ I_X\) and \(\pi_{Q'} \circ I_X\), maps \(P \to P'\) and \(P \to Q'\) respectively,
can be unpacked using the four "block" linear maps between the components of the two decompositions
\(V = P \oplus Q = P' \oplus Q'\):
\[
A = \pi_{P'}|_P : P \to P', \qquad
B = \pi_{Q'}|_P : P \to Q', \qquad
C = \pi_{P'}|_Q : Q \to P', \qquad
D = \pi_{Q'}|_Q : Q \to Q'.
\]
For \(v \in P\), the definition of \(I_X\) gives \(I_X(v) = v + Xv\) with \(v \in P\) and \(Xv \in Q\); applying
\(\pi_{P'}\) component by component,
\[
\pi_{P'} \circ I_X(v) = \pi_{P'}(v) + \pi_{P'}(Xv) = Av + C(Xv) = (A + CX)v,
\]
and analogously \(\pi_{Q'} \circ I_X(v) = (B + DX)v\). Substituting into the formula for \(X'\) yields the transition
formula
\[
X' = (B + DX)(A + CX)^{-1}.
\]
Smoothness in \(X\) is now a matter of reading this formula at the matrix level. Once bases for \(P, Q, P', Q'\) are
fixed, the four maps \(A, B, C, D\) are constant matrices (they depend only on the choice of decompositions, not on
\(X\)), and \(X\) itself is an \((n - k) \times k\) matrix. The entries of \(A + CX\) and \(B + DX\) are therefore
polynomial — in fact affine — in the entries of \(X\), hence smooth. The entries of \((A + CX)^{-1}\) are, by the
adjugate formula for matrix inverses,
rational functions of the entries of \(A + CX\), with denominator the polynomial \(\det(A + CX)\). On the overlap
\(\varphi_Q(U_Q \cap U_{Q'})\) the matrix \(A + CX\) is invertible (this is precisely the full-rank condition from
hypothesis (ii), so \(\det(A + CX) \ne 0\)), and consequently the entries of \((A + CX)^{-1}\) are smooth in the entries
of \(X\). The product \((B + DX)(A + CX)^{-1}\) is then smooth, and so is each entry of \(X'\). Hypothesis (iii) holds.
Hypothesis (iv): countably many charts cover \(G_k(V)\).
A stronger statement is true: finitely many charts cover \(G_k(V)\), which implies the countable cover demanded
by hypothesis (iv). To see this, fix a basis \((E_1, \ldots, E_n)\) for \(V\) and consider the finite collection of
partitions of the basis into a \(k\)-element subset and an \((n - k)\)-element subset. There are \(\binom{n}{k}\) such
partitions; each yields a complementary pair \((P, Q)\) where \(P\) is the span of the chosen \(k\) basis vectors and
\(Q\) is the span of the remaining \(n - k\). It remains to show that every \(k\)-dimensional subspace \(S \subseteq V\)
has trivial intersection with at least one such \(Q\).
Fix \(S \in G_k(V)\) and choose any basis \((e_1, \ldots, e_k)\) of \(S\). Expressing each \(e_i\) in the fixed basis
\((E_1, \ldots, E_n)\) gives a \(k \times n\) coefficient matrix of rank \(k\) (the rows are linearly independent because
the \(e_i\) are). A rank-\(k\) matrix has at least one \(k \times k\) submatrix that is invertible; the columns of that
submatrix correspond to a subset of \(k\) basis vectors of \(V\), say with index set \(I \subseteq \{1, \ldots, n\}\) of
size \(k\). Take \(P = \operatorname{span}\{E_i : i \in I\}\) and \(Q = \operatorname{span}\{E_j : j \notin I\}\). The
invertibility of the \(I\)-indexed submatrix says that the projection \(\pi_P|_S : S \to P\) is an isomorphism, which is
equivalent (as in the chart construction above) to \(S \cap Q = \{0\}\). Hence \(S \in U_Q\) for the partition indexed
by \(I\). Hypothesis (iv) holds.
Hypothesis (v): the separation property.
Given distinct subspaces \(S, S' \in G_k(V)\), we exhibit a single chart \(U_Q\) containing both, satisfying the first
clause of hypothesis (v). It suffices to find an \((n - k)\)-dimensional subspace \(Q \subseteq V\) with
\(S \cap Q = \{0\}\) and \(S' \cap Q = \{0\}\) simultaneously; then \(S, S' \in U_Q\).
Such a \(Q\) exists for any two \(k\)-dimensional subspaces, by the following construction. Consider the sum
\(S + S' \subseteq V\), of some dimension \(m\) with \(k \le m \le n\). Choose a basis of \(S + S'\) and extend it to a
basis of all of \(V\). The argument now proceeds by selecting a complement to \(S\) inside \(S + S'\) that also
complements \(S'\). Within the finite-dimensional space \(S + S'\), a subspace \(R\) of dimension \(m - k\) with
\(S \cap R = \{0\}\) and \(S' \cap R = \{0\}\) can be found directly: working in \(S + S'\), the set of vectors lying in
\(S\) or in \(S'\) is a union of two proper subspaces (proper because \(\dim S = \dim S' = k < m\) unless
\(S = S' = S + S'\), the excluded case \(S = S'\)), and over an infinite field the union of finitely many proper
subspaces never exhausts the whole space; one may therefore select \(m - k\) vectors of \(S + S'\), one at a time, each
chosen outside the span of the previously chosen vectors together with \(S\) and with \(S'\). The span \(R\) of these
\(m - k\) vectors then satisfies \(S \cap R = S' \cap R = \{0\}\) and \(S \oplus R = S' \oplus R = S + S'\). Finally,
set \(Q = R \oplus T\), where \(T\) is any complement of \(S + S'\) in \(V\) (so \(\dim T = n - m\)); then
\(\dim Q = (m - k) + (n - m) = n - k\), and \(Q\) meets both \(S\) and \(S'\) trivially because \(R\) does within
\(S + S'\) and \(T\) lies outside \(S + S'\) entirely. Hypothesis (v) holds.
The Grassmannian as a Smooth Manifold
All five hypotheses of the smooth manifold chart lemma have been verified for the candidate atlas \(\{(U_Q, \varphi_Q)\}\).
The lemma now delivers the smooth structure with no further work.
Theorem: The Grassmannian is a Smooth Manifold
Let \(V\) be an \(n\)-dimensional real vector space and \(0 \le k \le n\). The Grassmannian \(G_k(V)\) carries a
unique smooth manifold structure of dimension \(k(n - k)\) in which each chart \((U_Q, \varphi_Q)\), built from a
complementary pair \((P, Q)\) of subspaces of \(V\) as above, is a smooth chart.
Proof:
The candidate atlas \(\{(U_Q, \varphi_Q)\}\), indexed over all complementary pairs in \(V\), satisfies the five
hypotheses of the
smooth manifold chart lemma
by the verifications above: bijection onto an open subset of \(\mathbb{R}^{k(n-k)}\) (i), openness of chart-image
overlaps via the full-rank characterization (ii), smoothness of the transition formula
\(X' = (B + DX)(A + CX)^{-1}\) via the adjugate formula (iii), a finite cover by partition charts (iv), and the
separation property via a generic-complement argument (v). The lemma produces a unique smooth manifold structure on
\(G_k(V)\) of dimension \(\dim L(P; Q) = k(n - k)\) in which each \((U_Q, \varphi_Q)\) is a smooth chart.
\(\blacksquare\)
The smooth structure on \(\mathbb{RP}^n\) constructed on the
previous page
is the case \(k = 1\), \(V = \mathbb{R}^{n+1}\) of the present construction: a one-dimensional subspace of
\(\mathbb{R}^{n+1}\) is a line through the origin, and choosing a complement \(Q\) of dimension \(n\) makes \(U_Q\) the
set of lines not contained in \(Q\), parametrized by \(L(P; Q) \cong \mathbb{R}^n\). The two smooth structures agree:
the affine charts of the projective-space construction differ from the present graph charts only by a choice of basis,
and the change of basis is a smooth diffeomorphism between their image domains. As a sanity check, the dimension formula
\(k(n - k) = 1 \cdot n = n\) matches the dimension of \(\mathbb{RP}^n\) established there.
PCA and the Grassmannian
Real-valued data analysis frequently produces, as its essential output, not a vector but a subspace.
Principal-component analysis
is the prototype. Given a centered data matrix in \(\mathbb{R}^n\), PCA selects the
\(k\)-dimensional subspace of \(\mathbb{R}^n\) that captures maximal variance — the subspace onto which projection
best preserves the data, in the least-squares sense. The standard reportage of this output is a list of orthonormal
principal directions \(\mathbf{w}_1, \ldots, \mathbf{w}_k\), but that list is not unique: any orthogonal change of
basis within the span produces a different list with the same span and the same projection. The genuine PCA output
is a single point of the Grassmannian \(G_k(\mathbb{R}^n)\), and the optimization
\[
\underset{W \in G_k(\mathbb{R}^n)}{\arg\min} \sum_i \bigl\| \mathbf{x}_i - \operatorname{proj}_W \mathbf{x}_i \bigr\|^2
\]
that PCA implicitly solves is an optimization on the smooth manifold \(G_k(\mathbb{R}^n)\).
The same subspace-as-output picture recurs throughout applied geometry. Attention heads in transformer architectures
select context-dependent subspaces of representation space; subspace tracking in signal processing follows a smooth
path on a Grassmannian as new data arrives; the tangent spaces of low-dimensional data manifolds in the manifold
hypothesis are subspaces of the ambient feature space that vary smoothly with the base point. The smooth structure
constructed in this section is what makes "varying smoothly" a well-defined statement in each case, and it is what
allows gradient-based optimization to operate on subspace-valued quantities.