§ 4. Differential Manifolds
221
Manifolds have been defined using atlases, but the only thing that really
matters in a manifold X is the set C
r (U ) associated to all open subsets
U ⊂ X of numerical functions defined and of class C
r on U . As in the
case of the sphere, these functions are characterized by a local property: if
a function f is defined on an union U of open subsets U i , then f ∈ C
r (U )
if and only if the restriction of f to each U i belongs to C
r (U i ). Apart from
its atlas, there are many other useful local charts in a manifold X defined in
this manner, namely the charts of class C
r that, for short, will be henceforth
almost always called charts, or local charts, without further precision when
there is no ambiguity. These are the topological charts (U, ϕ) such that, for
any open subset U
⊂ U , the homeomorphism ϕ transforms C
r (U
) into the
set of C
r functions (in the usual sense) on the open subset ϕ(U
) of R
d . In
other words, a function defined on U
is of class C
r if and only if it is a C
r
function in the classical sense of the coordinates ξ
i = ϕ
i (x). In particular, the
coordinate functions ϕ
i (x) must be of class C
r on U , but choosing randomly
d functions in C
r (U ) is obviously not sufficient to obtain such a chart. Loosely
speaking, an open subset U of X will be said to be open Cartesian if it is the
domain of a chart (U, ϕ).
If (U, ϕ) and (V, ψ) are charts of class C
r of X, the C
r functions on U ∩ V
must be the same, whether they be given by ξ = ϕ(x) or η = ψ(x). As in the
above case of an atlas, this leads to the conclusion that, that the relations
η = θ(ξ), ξ = ρ(η) must hold in U ∩ V , i.e.
ϕ = ρ ◦ ψ , ψ = θ ◦ ϕ ,
(11.2)
where θ : ϕ(U ∩ V ) −→ ψ(U ∩ V ) and ρ : ψ(U ∩ V ) −→ ϕ(U ∩ V ) are of class
C
r , in other words are mutually inverse diffeomorphisms. The (formulas of)
change of chart (or coordinate) will be denoted ρ and θ. I do so because this
was the notation used by the inventors of absolute differential calculus (n
◦ 3,
(ii)), who without knowing it were working in the theory of manifolds.
(iii) Some Examples. Take E to be a d-dimensional Cartesian space and
choose a basis (a i ) for E. The point ϕ(x) = ξ
i e i ∈ R
d can be associated to
each x = ξ
i a i ∈ E. Its canonical coordinates are those of x with respect to
the basis of E. This gives a topological chart (E, ϕ) for E which, on its own,
is an atlas of E and so turns E into a manifold of class C
∞ . The ξ
i undergo
a linear transformation under a change of basis in E. Clearly, the manifold
structure thus defined on E does not depend on the choice of the basis (a i )
and for any open subset U ⊂ E, C
r (U ) is for all r a set of C
r functions on
U in the classical sense.
In the case of the unit sphere X discussed above, there are, among others,
two charts, a priori of class C
0 , using the stereographic projection from the
north or the south pole of X; they map the open complement U of this pole in
X to R
2 . If we consider the projection from the north pole, then the function
ϕ is easily seen to be given by
ϕ(x, y, z) = [x/(1 − z), y/(1 − z)]
221
Manifolds have been defined using atlases, but the only thing that really
matters in a manifold X is the set C
r (U ) associated to all open subsets
U ⊂ X of numerical functions defined and of class C
r on U . As in the
case of the sphere, these functions are characterized by a local property: if
a function f is defined on an union U of open subsets U i , then f ∈ C
r (U )
if and only if the restriction of f to each U i belongs to C
r (U i ). Apart from
its atlas, there are many other useful local charts in a manifold X defined in
this manner, namely the charts of class C
r that, for short, will be henceforth
almost always called charts, or local charts, without further precision when
there is no ambiguity. These are the topological charts (U, ϕ) such that, for
any open subset U
⊂ U , the homeomorphism ϕ transforms C
r (U
) into the
set of C
r functions (in the usual sense) on the open subset ϕ(U
) of R
d . In
other words, a function defined on U
is of class C
r if and only if it is a C
r
function in the classical sense of the coordinates ξ
i = ϕ
i (x). In particular, the
coordinate functions ϕ
i (x) must be of class C
r on U , but choosing randomly
d functions in C
r (U ) is obviously not sufficient to obtain such a chart. Loosely
speaking, an open subset U of X will be said to be open Cartesian if it is the
domain of a chart (U, ϕ).
If (U, ϕ) and (V, ψ) are charts of class C
r of X, the C
r functions on U ∩ V
must be the same, whether they be given by ξ = ϕ(x) or η = ψ(x). As in the
above case of an atlas, this leads to the conclusion that, that the relations
η = θ(ξ), ξ = ρ(η) must hold in U ∩ V , i.e.
ϕ = ρ ◦ ψ , ψ = θ ◦ ϕ ,
(11.2)
where θ : ϕ(U ∩ V ) −→ ψ(U ∩ V ) and ρ : ψ(U ∩ V ) −→ ϕ(U ∩ V ) are of class
C
r , in other words are mutually inverse diffeomorphisms. The (formulas of)
change of chart (or coordinate) will be denoted ρ and θ. I do so because this
was the notation used by the inventors of absolute differential calculus (n
◦ 3,
(ii)), who without knowing it were working in the theory of manifolds.
(iii) Some Examples. Take E to be a d-dimensional Cartesian space and
choose a basis (a i ) for E. The point ϕ(x) = ξ
i e i ∈ R
d can be associated to
each x = ξ
i a i ∈ E. Its canonical coordinates are those of x with respect to
the basis of E. This gives a topological chart (E, ϕ) for E which, on its own,
is an atlas of E and so turns E into a manifold of class C
∞ . The ξ
i undergo
a linear transformation under a change of basis in E. Clearly, the manifold
structure thus defined on E does not depend on the choice of the basis (a i )
and for any open subset U ⊂ E, C
r (U ) is for all r a set of C
r functions on
U in the classical sense.
In the case of the unit sphere X discussed above, there are, among others,
two charts, a priori of class C
0 , using the stereographic projection from the
north or the south pole of X; they map the open complement U of this pole in
X to R
2 . If we consider the projection from the north pole, then the function
ϕ is easily seen to be given by
ϕ(x, y, z) = [x/(1 − z), y/(1 − z)]