§ 4. Differential Manifolds
247
y −→
y
1 , . . . , y
p
from R
q = Y onto R
p induces a diffeomorphism from U onto an open subset
U
of R
p . The other coordinates x
j being of class C
r on U , we get the relations
y
j = f
j
y
1 , . . . , y
p
for all y ∈ U (p + 1 ≤ j ≤ q) ,
(13.10)
where f
j
∈ C
r (U
). Any y ∈ Y sufficiently near a so that (y
1 , . . . , y
p ) ∈ U
and satisfying relations (10) is then in U since there is a one (and only one)
x ∈ U whose first p coordinates are given in U
, its other q − p coordinates
satisfying (10). In other words,
y ∈ U ⇐⇒
y
1 , . . . , y
p
∈ U
& y
j = f
j
y
1 , . . . , y
p
.
(13.11)
If (f
p+1 , . . . , f
q ) is considered to be a map f from U
to R
q−p , this is equivalent to saying that in the neighbourhood of a, the submanifold X of R
q is
the graph of a map from R
p to R
q−p . Simply using coordinate changes given
by canonical permutations, precursors were right to believe that, locally, any
“ good ” curve or surface is the graph of a “ good ” function. As seen in section (i), this is in particular the case if X is defined globally by the “ implicit
equations ” F
k (y
1 , . . . , y
q ) = 0 provided the rank of the map y → (F
k (y)) is
constant in the neighbourhood of X or else X is the image of an open subset
of R
p under an open immersion.
Having said this, the elementary classical definition of a tangent vector
u to a submanifold X of Y at x, for example to a sphere, says that there
is a curve t → μ(t) in X such that μ(0) = x, μ
(0) = u, where, μ
(0) =
lim[μ(t) − μ(0)]/t here. The setT x (X) of these vectors is a vector subspace
65
of Y , not to be confused with the abstract vector space X
(x). Indeed, in
the neighbourhood of x, X may be supposed to the graph of a map f from
R
p to R
q−p . As R
q is identified to R
p
× R
q−p , x = (a, b) with a ∈ R
p and
b = f (a) ∈ R
q−p . Similarly, μ(t) = (μ 1 (t), μ 2 (t)) with μ 1 (t) ∈ R
p and
μ 2 (t) = f [μ 1 (t)] since μ(t) ∈ X. μ 1 can be arbitrarily chosen and, setting
k = μ
1 (0) = lim [μ 1 (t) − μ 1 (0)] /t, it follows that
u = μ
(0) = (k, f
(a)k) .
(13.12)
We, therefore, conclude that T x (X) is the image of R
p under the linear map
k → (k, f
(a)k), which gives the result. We thus recover the fact that, for
q = 2, p = 1, the slope of the tangent to the graph of the function f at
the point (a, f (a)) is the number f
(a) the linear map f
(a) of the general
case can be identified to in dimension 1. Observe also that, contrary to what
65 In classical geometry, the set Tx(X) thus defined is not the “ tangent plane ” to
X at x; depending on the point of view taken, it is a set either of points of Y
or of vectors with initial point x. Tx(X) is the vector subspace of Y which can
be deduced from the traditional tangent plane by using the translation mapping
the point x to 0.
247
y −→
y
1 , . . . , y
p
from R
q = Y onto R
p induces a diffeomorphism from U onto an open subset
U
of R
p . The other coordinates x
j being of class C
r on U , we get the relations
y
j = f
j
y
1 , . . . , y
p
for all y ∈ U (p + 1 ≤ j ≤ q) ,
(13.10)
where f
j
∈ C
r (U
). Any y ∈ Y sufficiently near a so that (y
1 , . . . , y
p ) ∈ U
and satisfying relations (10) is then in U since there is a one (and only one)
x ∈ U whose first p coordinates are given in U
, its other q − p coordinates
satisfying (10). In other words,
y ∈ U ⇐⇒
y
1 , . . . , y
p
∈ U
& y
j = f
j
y
1 , . . . , y
p
.
(13.11)
If (f
p+1 , . . . , f
q ) is considered to be a map f from U
to R
q−p , this is equivalent to saying that in the neighbourhood of a, the submanifold X of R
q is
the graph of a map from R
p to R
q−p . Simply using coordinate changes given
by canonical permutations, precursors were right to believe that, locally, any
“ good ” curve or surface is the graph of a “ good ” function. As seen in section (i), this is in particular the case if X is defined globally by the “ implicit
equations ” F
k (y
1 , . . . , y
q ) = 0 provided the rank of the map y → (F
k (y)) is
constant in the neighbourhood of X or else X is the image of an open subset
of R
p under an open immersion.
Having said this, the elementary classical definition of a tangent vector
u to a submanifold X of Y at x, for example to a sphere, says that there
is a curve t → μ(t) in X such that μ(0) = x, μ
(0) = u, where, μ
(0) =
lim[μ(t) − μ(0)]/t here. The setT x (X) of these vectors is a vector subspace
65
of Y , not to be confused with the abstract vector space X
(x). Indeed, in
the neighbourhood of x, X may be supposed to the graph of a map f from
R
p to R
q−p . As R
q is identified to R
p
× R
q−p , x = (a, b) with a ∈ R
p and
b = f (a) ∈ R
q−p . Similarly, μ(t) = (μ 1 (t), μ 2 (t)) with μ 1 (t) ∈ R
p and
μ 2 (t) = f [μ 1 (t)] since μ(t) ∈ X. μ 1 can be arbitrarily chosen and, setting
k = μ
1 (0) = lim [μ 1 (t) − μ 1 (0)] /t, it follows that
u = μ
(0) = (k, f
(a)k) .
(13.12)
We, therefore, conclude that T x (X) is the image of R
p under the linear map
k → (k, f
(a)k), which gives the result. We thus recover the fact that, for
q = 2, p = 1, the slope of the tangent to the graph of the function f at
the point (a, f (a)) is the number f
(a) the linear map f
(a) of the general
case can be identified to in dimension 1. Observe also that, contrary to what
65 In classical geometry, the set Tx(X) thus defined is not the “ tangent plane ” to
X at x; depending on the point of view taken, it is a set either of points of Y
or of vectors with initial point x. Tx(X) is the vector subspace of Y which can
be deduced from the traditional tangent plane by using the translation mapping
the point x to 0.