§ 4. Differential Manifolds
233
of X, Y and Z at a, b and c, in drawing the diagram of the nine maps
involved in the question : f , g and p, ϕ, ψ and π, and the maps F , G and P
expressing f , g and p in the charts considered, and in applying repeatedly the
definitions and the multivariate chain rule: an excellent exercise to understand
the mechanism of manifolds.
Because of the definition of f
(a), as in the classic situation (§ 1, n
◦ 2, (v)),
it is possible to define the rank of f at a : it is the dimension of the image
subspace Y
(b) of X
(a) under f
(a), i.e. the rank of the linear map f
(a) ;
if F expresses f in the local charts at a and b = f (a), the rank of f at a is
clearly equal to that of F at the point ξ corresponding to a. It cannot exceed
dim(X), nor dim(Y ) ; if it is equal to dim(X), then f
(a) is injective and we
have an immersion ; if it is equal to dim(Y ), f
(a) is surjective and we have
a submersion. In the neighbourhood of a point a, the rank of f is at least
equal to its rank at a, so that the rank of an immersion or of a submersion is
constant in the neighbourhood of a. Maps with this latter property are the
subimmersions. They are characterzed by Theorem 1 of § 1, n
◦ 3, (i), whose
generalization to manifolds is obvious.
(iv) Partial differentials. The tangent space Z
(a) at a point c = (a, b)
of a product manifold Z = X × Y is easily determined. Indeed, in this case,
there are projections pr 1 : Z −→ X and pr 2 : Z −→ Y given by (x, y) → x
and (x, y) → y ; so their tangent maps define maps Z
(c) −→ X
(a) and
Z
(c) −→ Y
(b), whence a map Z
(c) −→ X
(a) × Y
(b). As local charts
reduce the case to one where X and Y are open subsets of Cartesian spaces,
this map is clearly linear and bijective. We make no distinctions between
Z
(c) and X
(a) × Y
(b). If h ∈ Z
(c) is defined by a curve
t −→ μ(t) = (μ 1 (t), μ 2 (t)) ,
its images in X
(a) and Y
(b) are defined by the curves μ 1 and μ 2 .
Now, let X, Y, Z be three manifolds and f : X × Y −→ Z a homomorphism . We compute its tangent map at (a, b). If c = f (a, b), it maps
X
(a) × Y
(b) to Z
(c), and if h ∈ X
(a), k ∈ Y
(b) are defined by the paths
γ(t) and δ(t) with initial points a and b, the image of (h, k) is defined by the
path t → f [γ(t), δ(t)]. As (h, k) = (h, 0) + (0, k), it suffices to add the images
of (h, 0) and (0, k). The first one is defined by the path t → f [γ(t), b]. Thus
the tangent map to x → f (x, b), which will be denoted f
X (a, b) or d 1 f (a, b),
as well as the tangent map f
Y (a, b) or d 2 f (a, b) to y → f (a, y) need to be
considered. This implies that f
(a, b) is the map
f
(a, b) : (h, k) −→ f
X (a, b)h + f
Y (a, b)k .
(12.15)
The relation with the formulas of n
◦ 2, (iii), in particular (2.24), is clear ;
besides, thanks to local charts, (15) reduces to (2.24).
(v) The manifold of tangent vectors. Denote by X
or T (X) the set of
all tangent vectors to X, i.e of couples (x, h) with x ∈ X and h ∈ X
(x) . By
233
of X, Y and Z at a, b and c, in drawing the diagram of the nine maps
involved in the question : f , g and p, ϕ, ψ and π, and the maps F , G and P
expressing f , g and p in the charts considered, and in applying repeatedly the
definitions and the multivariate chain rule: an excellent exercise to understand
the mechanism of manifolds.
Because of the definition of f
(a), as in the classic situation (§ 1, n
◦ 2, (v)),
it is possible to define the rank of f at a : it is the dimension of the image
subspace Y
(b) of X
(a) under f
(a), i.e. the rank of the linear map f
(a) ;
if F expresses f in the local charts at a and b = f (a), the rank of f at a is
clearly equal to that of F at the point ξ corresponding to a. It cannot exceed
dim(X), nor dim(Y ) ; if it is equal to dim(X), then f
(a) is injective and we
have an immersion ; if it is equal to dim(Y ), f
(a) is surjective and we have
a submersion. In the neighbourhood of a point a, the rank of f is at least
equal to its rank at a, so that the rank of an immersion or of a submersion is
constant in the neighbourhood of a. Maps with this latter property are the
subimmersions. They are characterzed by Theorem 1 of § 1, n
◦ 3, (i), whose
generalization to manifolds is obvious.
(iv) Partial differentials. The tangent space Z
(a) at a point c = (a, b)
of a product manifold Z = X × Y is easily determined. Indeed, in this case,
there are projections pr 1 : Z −→ X and pr 2 : Z −→ Y given by (x, y) → x
and (x, y) → y ; so their tangent maps define maps Z
(c) −→ X
(a) and
Z
(c) −→ Y
(b), whence a map Z
(c) −→ X
(a) × Y
(b). As local charts
reduce the case to one where X and Y are open subsets of Cartesian spaces,
this map is clearly linear and bijective. We make no distinctions between
Z
(c) and X
(a) × Y
(b). If h ∈ Z
(c) is defined by a curve
t −→ μ(t) = (μ 1 (t), μ 2 (t)) ,
its images in X
(a) and Y
(b) are defined by the curves μ 1 and μ 2 .
Now, let X, Y, Z be three manifolds and f : X × Y −→ Z a homomorphism . We compute its tangent map at (a, b). If c = f (a, b), it maps
X
(a) × Y
(b) to Z
(c), and if h ∈ X
(a), k ∈ Y
(b) are defined by the paths
γ(t) and δ(t) with initial points a and b, the image of (h, k) is defined by the
path t → f [γ(t), δ(t)]. As (h, k) = (h, 0) + (0, k), it suffices to add the images
of (h, 0) and (0, k). The first one is defined by the path t → f [γ(t), b]. Thus
the tangent map to x → f (x, b), which will be denoted f
X (a, b) or d 1 f (a, b),
as well as the tangent map f
Y (a, b) or d 2 f (a, b) to y → f (a, y) need to be
considered. This implies that f
(a, b) is the map
f
(a, b) : (h, k) −→ f
X (a, b)h + f
Y (a, b)k .
(12.15)
The relation with the formulas of n
◦ 2, (iii), in particular (2.24), is clear ;
besides, thanks to local charts, (15) reduces to (2.24).
(v) The manifold of tangent vectors. Denote by X
or T (X) the set of
all tangent vectors to X, i.e of couples (x, h) with x ∈ X and h ∈ X
(x) . By