§ 4. Differential Manifolds
241
The most important case is that of an immersion or, as used to be said
in the past, a submanifold defined by a parametric representation, a variable
point of Z depending on the “ parameter ” x ∈ X. In the best of cases, f is
both an immersion from X to Y and a homeomorphism from X onto f (X) ;
f is then said to be an embedding of X in Y or, in obsolete language, a
parametric eigenspace representation of Z . To suppose that f is an open and
injective immersion would be equivalent since f
−1 is then continuous. As,
for any sufficiently small open subset U ⊂ X, f is a diffeomorphism from
U onto the open subset f (U ) of the manifold f (X) and as f is a global
homeomorphism from X onto f (X), it is in fact a diffeomorphism from X
onto f (X).
H. Whitney showed that, given some inoffensive countability assumptions,
any n-dimensional manifold admits an embedding into R
2n . This well-known
theorem is hard to prove – easy theorems rarely become famous – except
in the fairly elementary case of compact manifolds;
58 even Dieudonn´ e, who
proves it for R
2n+1 (XVI.25, exercises 2 and 13) instead of R
2n , retreated
before the complete result. Doubts may arise as to the practical usefulness
of this type of theorem since a useful embedding of a manifold into a Cartesian space is generally one that can be explicitly constructed from the specific data of the situation. If, for example, the universe happened to be a
4-dimensional “ curved ” manifold, looking for an artificial embedding of it
into a 8-dimensional Cartesian space would not be very useful, though with
physicists. . .
58 If X is compact of dimension d, it can be covered by finitely many N charts
(Up, ϕp) such that for all p, ϕp(Up) is the cube |ξ
i | < 2 of R
d ; denoting by Vp
the set of x ∈ Up for which ϕp(x) is in the cube |ξ
i | < 1, the Vp may be assumed
to cover X. Now, there is a C
∞ function h on R
d equal to 1 for |ξ
i | < 1 and
to 0 for |ξ
i | > 3/2 (for d = 1, see Chapter V, n
◦ 29 ; the general case follows in
an obvious manner). Replacing the Up by the Vp and the ϕp by the restrictions
to Vp of the functions h[ϕp(x)], X can be covered by the N charts (Vp, ϕp) for
which the ϕp (extended by 0 outside Up) are defined and of class C
r on all of X.
The map
x −→ (ϕ1(x), . . . , ϕN (x))
from X to R
d ×. . .×R
d = R
Nd then has rank d everywhere, but is not necessarily
injective. To obtain a homeomorphism, use a partition of unity, i.e. a family of
functions (θp) on X satisfying
θp(x) = 1 for all x and whose supports are
contained in the Vp. The map
x −→ (ϕ1(x), . . . , ϕN (x), θ1(x), . . . , θN (x))
is then continuous and injective, hence a homeomorphism from X onto its image
since X is compact, and has rank d everywhere. This gives an embedding of X
into R
N (d+1) .
241
The most important case is that of an immersion or, as used to be said
in the past, a submanifold defined by a parametric representation, a variable
point of Z depending on the “ parameter ” x ∈ X. In the best of cases, f is
both an immersion from X to Y and a homeomorphism from X onto f (X) ;
f is then said to be an embedding of X in Y or, in obsolete language, a
parametric eigenspace representation of Z . To suppose that f is an open and
injective immersion would be equivalent since f
−1 is then continuous. As,
for any sufficiently small open subset U ⊂ X, f is a diffeomorphism from
U onto the open subset f (U ) of the manifold f (X) and as f is a global
homeomorphism from X onto f (X), it is in fact a diffeomorphism from X
onto f (X).
H. Whitney showed that, given some inoffensive countability assumptions,
any n-dimensional manifold admits an embedding into R
2n . This well-known
theorem is hard to prove – easy theorems rarely become famous – except
in the fairly elementary case of compact manifolds;
58 even Dieudonn´ e, who
proves it for R
2n+1 (XVI.25, exercises 2 and 13) instead of R
2n , retreated
before the complete result. Doubts may arise as to the practical usefulness
of this type of theorem since a useful embedding of a manifold into a Cartesian space is generally one that can be explicitly constructed from the specific data of the situation. If, for example, the universe happened to be a
4-dimensional “ curved ” manifold, looking for an artificial embedding of it
into a 8-dimensional Cartesian space would not be very useful, though with
physicists. . .
58 If X is compact of dimension d, it can be covered by finitely many N charts
(Up, ϕp) such that for all p, ϕp(Up) is the cube |ξ
i | < 2 of R
d ; denoting by Vp
the set of x ∈ Up for which ϕp(x) is in the cube |ξ
i | < 1, the Vp may be assumed
to cover X. Now, there is a C
∞ function h on R
d equal to 1 for |ξ
i | < 1 and
to 0 for |ξ
i | > 3/2 (for d = 1, see Chapter V, n
◦ 29 ; the general case follows in
an obvious manner). Replacing the Up by the Vp and the ϕp by the restrictions
to Vp of the functions h[ϕp(x)], X can be covered by the N charts (Vp, ϕp) for
which the ϕp (extended by 0 outside Up) are defined and of class C
r on all of X.
The map
x −→ (ϕ1(x), . . . , ϕN (x))
from X to R
d ×. . .×R
d = R
Nd then has rank d everywhere, but is not necessarily
injective. To obtain a homeomorphism, use a partition of unity, i.e. a family of
functions (θp) on X satisfying
θp(x) = 1 for all x and whose supports are
contained in the Vp. The map
x −→ (ϕ1(x), . . . , ϕN (x), θ1(x), . . . , θN (x))
is then continuous and injective, hence a homeomorphism from X onto its image
since X is compact, and has rank d everywhere. This gives an embedding of X
into R
N (d+1) .