§ 4. Differential Manifolds
223
in U i ∩ U j are C
∞ (or even rational) functions of the coordinates ϕ
j (D)
and conversely. The result follows, and this atlas defines a C
∞ manifold
structure on P n (R). C
r functions on an open subset U ⊂ P n (R) are just
the homogeneous functions of class C
r (in the usual sense) on the open set
p
−1 (U ).
Exercise. Check Hausdorff’s axiom.
Replacing the 1-dimensional subspaces in the previous construction by
p-dimensional ones for given p, it is possible to generalize. But it is slightly
less simple and we leave it to the reader to give detailed proofs. Let E be
an n-dimensional Cartesian space and Ω ⊂ E
p the set of sequences x =
(x 1 , . . . , x p ) of p linearly independent vectors in E; it is an open subset of
E
p . Any p-dimensional subspace H of E is generated by the “ components ”
x i of some x ∈ Ω and x, y ∈ Ω generate the same subspaces if and only if
there is a matrix (g
j
i ) ∈ GL p (R) such that y i = g
j
i x j . This is an equivalence
relation, so that the set X = G p (E) of p-dimensional subspaces of E is the
quotient of Ω modulo this relation. If p : Ω −→ X denotes the obvious map,
we then get a topology on X as in the case p = 1 : U ⊂ X is open if and only
if so is p
−1 (U ) in Ω (or E
p ). Having done this, if U ⊂ X is open, C
∞ (U ) is,
by definition, the set of functions f for which f ◦ p is C
∞ on the open subset
p
−1 (U ) of Ω. There are verifications to be done and charts to be found; we
proceed as follows.
For this, choose a (n − p)-dimensional vector subspace F in E and let X F
denote the set of H ∈ X such that H ∩ F = {0} or, what amounts to the
same for reasons of dimension, such that E = F ⊕ H, a direct sum. It is not
hard to see that X F is open in X. Then choose p vectors a i ∈ E generating
a subspace H 0 such that E = F ⊕ H 0 . If H ∈ X F , the relation E = F ⊕ H
shows that, for all i, there is a unique x i ∈ F such that a i − x i ∈ H. Setting
ϕ(H) = (x 1 , . . . , x p ) for all H ∈ X F , define a bijection from X F onto F
p – a
linear algebra exercise – and hence a chart (X F , ϕ) for X F , a priori purely
set-theoretical.
Couples (X F , ϕ) depending on the choice of F and of vectors a i are in
fact topological charts pairwise C
∞ -compatibles and they define the p(n−p)dimensional manifold structure of the grassmannian X = G p (E).
The latter is compact . To see this, choose a Hilbert or Euclidean scalar
product (x|y) on E and note that in all subspaces H of E, there are bases
x = (x 1 , . . . , x p ), orthonormal with respect to it: (x i |x j ) = 1 or 0 . As these
relations are conserved when passing to the limit and prove that the x i are
linearly independent, the set Ω 0 ⊂ Ω of these systems is closed in E
p . It is
also compact since it is bounded. As p : Ω −→ X is continuous and maps Ω 0
onto X, the compactness of X is obvious.
49
Real or complex (replace R by C in what precedes) Grassmannians were
invented in the 19th century in order to generalize of projective geometry
and its real or imaginary “ points at infinity ”. In former times, they used to
49 For further information and other methods, see Dieudonn´ e, XVI.11.
223
in U i ∩ U j are C
∞ (or even rational) functions of the coordinates ϕ
j (D)
and conversely. The result follows, and this atlas defines a C
∞ manifold
structure on P n (R). C
r functions on an open subset U ⊂ P n (R) are just
the homogeneous functions of class C
r (in the usual sense) on the open set
p
−1 (U ).
Exercise. Check Hausdorff’s axiom.
Replacing the 1-dimensional subspaces in the previous construction by
p-dimensional ones for given p, it is possible to generalize. But it is slightly
less simple and we leave it to the reader to give detailed proofs. Let E be
an n-dimensional Cartesian space and Ω ⊂ E
p the set of sequences x =
(x 1 , . . . , x p ) of p linearly independent vectors in E; it is an open subset of
E
p . Any p-dimensional subspace H of E is generated by the “ components ”
x i of some x ∈ Ω and x, y ∈ Ω generate the same subspaces if and only if
there is a matrix (g
j
i ) ∈ GL p (R) such that y i = g
j
i x j . This is an equivalence
relation, so that the set X = G p (E) of p-dimensional subspaces of E is the
quotient of Ω modulo this relation. If p : Ω −→ X denotes the obvious map,
we then get a topology on X as in the case p = 1 : U ⊂ X is open if and only
if so is p
−1 (U ) in Ω (or E
p ). Having done this, if U ⊂ X is open, C
∞ (U ) is,
by definition, the set of functions f for which f ◦ p is C
∞ on the open subset
p
−1 (U ) of Ω. There are verifications to be done and charts to be found; we
proceed as follows.
For this, choose a (n − p)-dimensional vector subspace F in E and let X F
denote the set of H ∈ X such that H ∩ F = {0} or, what amounts to the
same for reasons of dimension, such that E = F ⊕ H, a direct sum. It is not
hard to see that X F is open in X. Then choose p vectors a i ∈ E generating
a subspace H 0 such that E = F ⊕ H 0 . If H ∈ X F , the relation E = F ⊕ H
shows that, for all i, there is a unique x i ∈ F such that a i − x i ∈ H. Setting
ϕ(H) = (x 1 , . . . , x p ) for all H ∈ X F , define a bijection from X F onto F
p – a
linear algebra exercise – and hence a chart (X F , ϕ) for X F , a priori purely
set-theoretical.
Couples (X F , ϕ) depending on the choice of F and of vectors a i are in
fact topological charts pairwise C
∞ -compatibles and they define the p(n−p)dimensional manifold structure of the grassmannian X = G p (E).
The latter is compact . To see this, choose a Hilbert or Euclidean scalar
product (x|y) on E and note that in all subspaces H of E, there are bases
x = (x 1 , . . . , x p ), orthonormal with respect to it: (x i |x j ) = 1 or 0 . As these
relations are conserved when passing to the limit and prove that the x i are
linearly independent, the set Ω 0 ⊂ Ω of these systems is closed in E
p . It is
also compact since it is bounded. As p : Ω −→ X is continuous and maps Ω 0
onto X, the compactness of X is obvious.
49
Real or complex (replace R by C in what precedes) Grassmannians were
invented in the 19th century in order to generalize of projective geometry
and its real or imaginary “ points at infinity ”. In former times, they used to
49 For further information and other methods, see Dieudonn´ e, XVI.11.