236
IX – Multivariate Differential and Integral Calculus
century by using Cartesian coordinates; the theorems used are, except for
generality and language, known since the 19th century. Since the 18th, people
have studied “ curves ” and “ surfaces ” in the plane or the space, sometimes
defined by an equation y = f (x) in the plane (the parabola y = x
2 for
example), sometimes by a parametric representation (the ellipse x = a. cos t,
y = b. sin t for example), sometimes by a relation between coordinates (the
sphere x
2 + y
2 + z
2 = 1 for example). More complicated curves and surfaces
have quickly been considered, for example the planar strophoid with equation
(x − 1)y
2 + x
2 (x + 1) = 0 ,
which can also be obtained by the parametric representation
x =
t
2
− 1
/
t
2 + 1
, y = tx ;
there are two arcs of simple curves in the neighbourhood of the origin meeting
at 0, with distinct tangents; this is an example of a singularity which does
not fall within the framework about to be defined (fig. 7).
The three classical methods recalled above fall within a more general
pattern: take two manifolds X and Y and a map f : X −→ Y and consider
either the image f (X) ⊂ Y (the case of parametric representations), or, for
some b ∈ Y , the set of solutions of f (x) = b in X (submanifold defined by
relations between coordinates). The graph of a function X −→ Y falls within
this general pattern either by considering it the image of X under the map
x → (x, f (x)) from X to X × Y , either as the set of solutions of y − f (x) = 0.
In this n
◦ , we are going to define submanifolds and then show how some may
be obtained by these methods – direct image of a map, inverse image of a
point under a map – provided we restrict ourselves to subimmersions, i.e. to
maps of constant rank.
(i) Submanifolds. Let X be a subset of a manifold Y of class C
r and
dimension q. For any open subset U of X, let C
r (U ) be the set of functions f
defined on U and satisfying the following property: for all a ∈ U , there is
an open neighbourhood V (a) of a in Y and a C
r function on V (a) which is
equal to f on U ∩ V (a). The example of the sphere (n
◦ 11, (i)) suggests that
X should be called a submanifold of Y if and only if the topological space
X has the structure of a manifold of a priori arbitrary dimension p, and for
which the C
r functions are precisely those that have been defined.
A first consequence of this definition is that the identity map x → x from
X to Y is then of class C
r : it follows from the definition of these maps. In
fact, it is an immersion, and so p ≤ q as expected since there is then a chart
(V, ψ) of Y at any a ∈ X – for convenience, it can always be supposed to
be cubic – such that ψ(X ∩ V ) is the face of the cube ψ(V ) defined by the
relations ξ
p+1 = . . . = ξ
q = 0, i.e. such that
ψ(V ∩ X) = ψ(V ) ∩ R
p ,
(13.1)
IX – Multivariate Differential and Integral Calculus
century by using Cartesian coordinates; the theorems used are, except for
generality and language, known since the 19th century. Since the 18th, people
have studied “ curves ” and “ surfaces ” in the plane or the space, sometimes
defined by an equation y = f (x) in the plane (the parabola y = x
2 for
example), sometimes by a parametric representation (the ellipse x = a. cos t,
y = b. sin t for example), sometimes by a relation between coordinates (the
sphere x
2 + y
2 + z
2 = 1 for example). More complicated curves and surfaces
have quickly been considered, for example the planar strophoid with equation
(x − 1)y
2 + x
2 (x + 1) = 0 ,
which can also be obtained by the parametric representation
x =
t
2
− 1
/
t
2 + 1
, y = tx ;
there are two arcs of simple curves in the neighbourhood of the origin meeting
at 0, with distinct tangents; this is an example of a singularity which does
not fall within the framework about to be defined (fig. 7).
The three classical methods recalled above fall within a more general
pattern: take two manifolds X and Y and a map f : X −→ Y and consider
either the image f (X) ⊂ Y (the case of parametric representations), or, for
some b ∈ Y , the set of solutions of f (x) = b in X (submanifold defined by
relations between coordinates). The graph of a function X −→ Y falls within
this general pattern either by considering it the image of X under the map
x → (x, f (x)) from X to X × Y , either as the set of solutions of y − f (x) = 0.
In this n
◦ , we are going to define submanifolds and then show how some may
be obtained by these methods – direct image of a map, inverse image of a
point under a map – provided we restrict ourselves to subimmersions, i.e. to
maps of constant rank.
(i) Submanifolds. Let X be a subset of a manifold Y of class C
r and
dimension q. For any open subset U of X, let C
r (U ) be the set of functions f
defined on U and satisfying the following property: for all a ∈ U , there is
an open neighbourhood V (a) of a in Y and a C
r function on V (a) which is
equal to f on U ∩ V (a). The example of the sphere (n
◦ 11, (i)) suggests that
X should be called a submanifold of Y if and only if the topological space
X has the structure of a manifold of a priori arbitrary dimension p, and for
which the C
r functions are precisely those that have been defined.
A first consequence of this definition is that the identity map x → x from
X to Y is then of class C
r : it follows from the definition of these maps. In
fact, it is an immersion, and so p ≤ q as expected since there is then a chart
(V, ψ) of Y at any a ∈ X – for convenience, it can always be supposed to
be cubic – such that ψ(X ∩ V ) is the face of the cube ψ(V ) defined by the
relations ξ
p+1 = . . . = ξ
q = 0, i.e. such that
ψ(V ∩ X) = ψ(V ) ∩ R
p ,
(13.1)