246
IX – Multivariate Differential and Integral Calculus
their images. Such a couple (X, f ) is sometimes called an immersed manifold
in Y , a notion that should not be confused with that of an embedding defined
above. The image f (X) is not generally a submanifold; it is a subset Z
equipped with a manifold structure such that the map id : Z −→ Y is an
immersion. This is in particular encountered in the theory of Lie groups.
The theory of differential equations contains many such phenomena. The
movements of a gyroscope turning at constant velocity around an axis one
of whose endpoints is fixed can be periodic, but it is an exceptional case.
In general, the free endpoint of its axis, whose angle with the vertical varies
between two limits determined by its initial velocity, describes a trajectory
that is everywhere dense in the part S of the sphere contained between these
two limit inclinations. It passes infinitely many times in the neighbourhood
of every point of S.
The problem detailed in this section can be generalized by considering
the map
t −→ (e (a 1 t) , . . . , e (a n t))
from R to T
n . Its image is everywhere dense in T
n if an only if the a i ∈ R
are linearly independent
64 over Q, i.e. if the relation
x 1 a 1 + . . . + x n a n = 0
x 1 , . . . , x n ∈ Q
implies that x i = 0 for all i. The proof is the same, but slightly less easy since
it first requires finding all the closed subgroups of R
n : such a subgroup is the
set of vectors which, with respect to a conveniently chosen basis (u i ) for R
n
can be written as
x
i u i , where x
i
∈ R for 1 ≤ i ≤ p, x
i
∈ Z for p +1 ≤ i ≤ q
and x
i = 0 for i ≥ q. Let us now return to more general manifolds.
(iv) Submanifolds of a Cartesian space : tangent vectors. In the case of a
p-dimensional submanifold X of a q-dimensional Cartesian space Y , a more
concrete description of tangent spaces X
(a) can be given ; it has no use – as
remarked with reason by Dieudonn´ e (El´ ements d’analyse, vol. 3, p. 2), it may
even lead the reader on a wrong track – but it makes it possible to relate
“ abstract ” tangent spaces X
(x) to “ tangent planes ” of classical geometry.
In what follows, we suppose that Y = R
q , and we will identify R
q to the
Cartesian product R
p
× R
q−p .
First note that, in this case, the 18th century method for representing a
plane curve either by an equation y = f (x), or by an equation x = g(y), can
be generalized here. Indeed if, as a general rule, we denote by y
j the canonical
coordinates of some y ∈ Y , we know that (end of (i)) in the neighbourhood of
all a ∈ X, the restrictions x
j of these q functions to X form a system of rank
p. Hence, up a permutation of the canonical coordinates, the first p functions
x
i may be supposed to be defined by a chart of X in the neighbourhood U
of a in X, which means that the projection
64 R is an infinite-dimensional vector space over Q.
IX – Multivariate Differential and Integral Calculus
their images. Such a couple (X, f ) is sometimes called an immersed manifold
in Y , a notion that should not be confused with that of an embedding defined
above. The image f (X) is not generally a submanifold; it is a subset Z
equipped with a manifold structure such that the map id : Z −→ Y is an
immersion. This is in particular encountered in the theory of Lie groups.
The theory of differential equations contains many such phenomena. The
movements of a gyroscope turning at constant velocity around an axis one
of whose endpoints is fixed can be periodic, but it is an exceptional case.
In general, the free endpoint of its axis, whose angle with the vertical varies
between two limits determined by its initial velocity, describes a trajectory
that is everywhere dense in the part S of the sphere contained between these
two limit inclinations. It passes infinitely many times in the neighbourhood
of every point of S.
The problem detailed in this section can be generalized by considering
the map
t −→ (e (a 1 t) , . . . , e (a n t))
from R to T
n . Its image is everywhere dense in T
n if an only if the a i ∈ R
are linearly independent
64 over Q, i.e. if the relation
x 1 a 1 + . . . + x n a n = 0
x 1 , . . . , x n ∈ Q
implies that x i = 0 for all i. The proof is the same, but slightly less easy since
it first requires finding all the closed subgroups of R
n : such a subgroup is the
set of vectors which, with respect to a conveniently chosen basis (u i ) for R
n
can be written as
x
i u i , where x
i
∈ R for 1 ≤ i ≤ p, x
i
∈ Z for p +1 ≤ i ≤ q
and x
i = 0 for i ≥ q. Let us now return to more general manifolds.
(iv) Submanifolds of a Cartesian space : tangent vectors. In the case of a
p-dimensional submanifold X of a q-dimensional Cartesian space Y , a more
concrete description of tangent spaces X
(a) can be given ; it has no use – as
remarked with reason by Dieudonn´ e (El´ ements d’analyse, vol. 3, p. 2), it may
even lead the reader on a wrong track – but it makes it possible to relate
“ abstract ” tangent spaces X
(x) to “ tangent planes ” of classical geometry.
In what follows, we suppose that Y = R
q , and we will identify R
q to the
Cartesian product R
p
× R
q−p .
First note that, in this case, the 18th century method for representing a
plane curve either by an equation y = f (x), or by an equation x = g(y), can
be generalized here. Indeed if, as a general rule, we denote by y
j the canonical
coordinates of some y ∈ Y , we know that (end of (i)) in the neighbourhood of
all a ∈ X, the restrictions x
j of these q functions to X form a system of rank
p. Hence, up a permutation of the canonical coordinates, the first p functions
x
i may be supposed to be defined by a chart of X in the neighbourhood U
of a in X, which means that the projection
64 R is an infinite-dimensional vector space over Q.