238
IX – Multivariate Differential and Integral Calculus
first p relations of (2) then show that the functions ϕ
i (1 ≤ i ≤ p) are the
restrictions to U of the first p functions ψ
j , and the following ones show that
ϕ(U ) = ψ(V ) ∩ R
p , which gives (1).
Conversely, if for all a ∈ X there is a chart (V, ψ) of Y satisfying (1),
then X is a submanifold of Y . Indeed, condition (1) shows that for any open
subset U of V ∩X, the f ∈ C
r (U ), where C
r (U ) is defined for all open subsets
of X like at the beginning of this section, are the C
r functions of the first p
coordinates ψ
i (x) on the open subset ψ(U ) of R
p ; denoting by ϕ
i
∈ C
r (U )
the restrictions to U of these first p functions ψ
i , we get a chart (U, ϕ) for X.
The charts of X obtained in this manner are pairwise C
r -compatibles since
so are the charts of Y used to construct them. Hence the result.
A corollary is that any submanifold X of a manifold Y is open in its
closure ¯
X, which means that ¯
X − X is closed in Y , or that a sequence of
points a n ∈ ¯
X converges to some a ∈ X only if a n ∈ X for large n. Indeed,
let (V, ψ) be a chart for Y at a such that V ∩ X is defined by relation (1) ; the
a n are in V for large n and so are the limits of points of X ∩ V ; as (1) shows
that ψ(V ∩ X) is closed in ψ(V ), ψ(a n ) ∈ ψ(V ∩ X), and so a n ∈ V ∩ X ⊂ X,
qed.
It goes without saying that if a chart (V, ψ) of Y at a ∈ X is chosen
randomly, the restrictions of the ψ
j to U = X ∩ V do not constitute a chart
for X; to start with, there are too many of them. But if X and Y have
dimensions p and q, p functions forming a chart for X can be extracted from
these q restrictions. Indeed, as shown above, there is a chart (U, ϕ) for X – it
may be assumed to be defined on U by choosing V sufficiently small – such
that the ϕ
i are the restrictions to U of the first p coordinate functions of a
chart (V, θ) of Y , which can here too be supposed to be defined on V . The θ
k
being the C
r functions on V of the ψ
j , the ϕ
i , are C
r functions on U of the
restrictions to U of the ψ
j ; but since the ϕ
i form a chart for X in U , these
restrictions are also C
r functions of the ϕ
i . The p × q Jacobian matrix of the
restrictions of the ψ
j with respect to the ϕ
i is, therefore, of maximum rank
p, and hence extracting a non-zero determinant of order p from it gives the
p functions sought.
More generally, if X is a p-dimensional manifold and if, in the neighbourhood of some a ∈ X, there are p C
r functions with non-zero Jacobian in
a local (hence in all) chart, these p functions define a chart for X in the
neighbourhood of a: this is the local inversion theorem.
Finally, note that, if X is a p-dimensional submanifold of a q-dimensional
manifold Y , by the canonical immersion from X to Y , for any x ∈ X,
the tangent space X
(x) can be identified with its image in Y
(x) under the
linear map id
(x) : X
(x) −→ Y
(x). In (ii) we will see how to determine this
subspace of Y
(x) using local equations of X.
Exercise 1. Let Y be a q-dimensional manifold and X a subset of Y
equipped with a manifold structure such that the map x → x is an immersion.
Show that X is a submanifold of Y .
IX – Multivariate Differential and Integral Calculus
first p relations of (2) then show that the functions ϕ
i (1 ≤ i ≤ p) are the
restrictions to U of the first p functions ψ
j , and the following ones show that
ϕ(U ) = ψ(V ) ∩ R
p , which gives (1).
Conversely, if for all a ∈ X there is a chart (V, ψ) of Y satisfying (1),
then X is a submanifold of Y . Indeed, condition (1) shows that for any open
subset U of V ∩X, the f ∈ C
r (U ), where C
r (U ) is defined for all open subsets
of X like at the beginning of this section, are the C
r functions of the first p
coordinates ψ
i (x) on the open subset ψ(U ) of R
p ; denoting by ϕ
i
∈ C
r (U )
the restrictions to U of these first p functions ψ
i , we get a chart (U, ϕ) for X.
The charts of X obtained in this manner are pairwise C
r -compatibles since
so are the charts of Y used to construct them. Hence the result.
A corollary is that any submanifold X of a manifold Y is open in its
closure ¯
X, which means that ¯
X − X is closed in Y , or that a sequence of
points a n ∈ ¯
X converges to some a ∈ X only if a n ∈ X for large n. Indeed,
let (V, ψ) be a chart for Y at a such that V ∩ X is defined by relation (1) ; the
a n are in V for large n and so are the limits of points of X ∩ V ; as (1) shows
that ψ(V ∩ X) is closed in ψ(V ), ψ(a n ) ∈ ψ(V ∩ X), and so a n ∈ V ∩ X ⊂ X,
qed.
It goes without saying that if a chart (V, ψ) of Y at a ∈ X is chosen
randomly, the restrictions of the ψ
j to U = X ∩ V do not constitute a chart
for X; to start with, there are too many of them. But if X and Y have
dimensions p and q, p functions forming a chart for X can be extracted from
these q restrictions. Indeed, as shown above, there is a chart (U, ϕ) for X – it
may be assumed to be defined on U by choosing V sufficiently small – such
that the ϕ
i are the restrictions to U of the first p coordinate functions of a
chart (V, θ) of Y , which can here too be supposed to be defined on V . The θ
k
being the C
r functions on V of the ψ
j , the ϕ
i , are C
r functions on U of the
restrictions to U of the ψ
j ; but since the ϕ
i form a chart for X in U , these
restrictions are also C
r functions of the ϕ
i . The p × q Jacobian matrix of the
restrictions of the ψ
j with respect to the ϕ
i is, therefore, of maximum rank
p, and hence extracting a non-zero determinant of order p from it gives the
p functions sought.
More generally, if X is a p-dimensional manifold and if, in the neighbourhood of some a ∈ X, there are p C
r functions with non-zero Jacobian in
a local (hence in all) chart, these p functions define a chart for X in the
neighbourhood of a: this is the local inversion theorem.
Finally, note that, if X is a p-dimensional submanifold of a q-dimensional
manifold Y , by the canonical immersion from X to Y , for any x ∈ X,
the tangent space X
(x) can be identified with its image in Y
(x) under the
linear map id
(x) : X
(x) −→ Y
(x). In (ii) we will see how to determine this
subspace of Y
(x) using local equations of X.
Exercise 1. Let Y be a q-dimensional manifold and X a subset of Y
equipped with a manifold structure such that the map x → x is an immersion.
Show that X is a submanifold of Y .