§ 4. Differential Manifolds
219
neighbourhood of (a, b, c) are, by definition, those that can be expressed in a
differentiable manner using its coordinates x, z. Etc. Thus, X can be written
as the union of six open subsets. On each of them there is a special homeomorphism onto an open subset of R
2 . Each of these homeomorphisms makes
it possible to give a reasonable definition of C
r (r ≤ ∞) functions on the
corresponding open subset of X.
This definition would, however, be insignificant if it gave incompatible
definitions of differentiability on the intersections of these open subsets; it
is not at all so. For example, take an open set U , where both {z > 0} and
{y < 0} hold ; using the hemisphere z > 0, the relation f ∈ C
r (U ) means
that f is a C
r function of (x, y) ; using the hemisphere y < 0, it means that
f is a C
r function of (x, z) . Hence it suffices to show that (x, y) is a C
∞
function of (x, z) on the open subset {z > 0} ∩ {y < 0} of X, and conversely.
This is obvious since
y = −
1 − x
2
− z
2
1/2 and z =
1 − x
2
− y
2
1/2
with 1 − x
2
− z
2 > 0 and 1 − x
2
− y
2 > 0.
This leads to a coherent definition of functions of class C
r on a random
open subset of the sphere.
It can also be formulated more directly. First, given a C
r function of
(x, y, z) on an open subset V of R
3 , it restriction to U = V ∩ X is of class C
r
in the previous sense. Conversely, consider a C
r function f on an open subset
U of X and take a neighbourhood of a point (a, b, c) ∈ U . If, for example
c > 0, the definition shows that in the neighbourhood of (a, b, c), f is a C
r
function of (x, y) defined in the neighbourhood of a point (a, b) of R
2 . The
composition of f and the projection (x, y, z) → (x, y) of R
3 onto R
2 is a C
r
function of (x, y, z) on a vertical cylinder having an open subset of the plane
(x, y) as base. The restriction of this function to X is equal to the given
function f on a neighbourhood of (a, b, c). In conclusion, a fuction f defined
on an open subset U of X is of class C
r if and only if, in the neighbourhood
of every point of U , it has a C
r extension on an open subset of R
3 .
(ii) The notion of a manifold of class C
r and dimension d is obtained by
generalizing the construction of C
r functions on the sphere.
To start with, a manifold X is a separated topological space. Hence there
is a category of sets in X called open and satisfying the two obvious conditions
(any union of open sets is open, the intersection of a finite number of open
sets is open), as well as Hausdorff’s condition: if a and b are two distinct
points of X, there are open disjoint subsets U and V containing a and b.
Topological spaces are the natural realm of the notion of continuity: a map
f : X −→ Y is continuous if and only if the inverse image f
−1 (V ) of any
open subset V of Y is an open subset of X.
A differential manifold X must also be a locally Cartesian topological
space: for each a ∈ X, there must be an open neighbourhood U homeomor-
219
neighbourhood of (a, b, c) are, by definition, those that can be expressed in a
differentiable manner using its coordinates x, z. Etc. Thus, X can be written
as the union of six open subsets. On each of them there is a special homeomorphism onto an open subset of R
2 . Each of these homeomorphisms makes
it possible to give a reasonable definition of C
r (r ≤ ∞) functions on the
corresponding open subset of X.
This definition would, however, be insignificant if it gave incompatible
definitions of differentiability on the intersections of these open subsets; it
is not at all so. For example, take an open set U , where both {z > 0} and
{y < 0} hold ; using the hemisphere z > 0, the relation f ∈ C
r (U ) means
that f is a C
r function of (x, y) ; using the hemisphere y < 0, it means that
f is a C
r function of (x, z) . Hence it suffices to show that (x, y) is a C
∞
function of (x, z) on the open subset {z > 0} ∩ {y < 0} of X, and conversely.
This is obvious since
y = −
1 − x
2
− z
2
1/2 and z =
1 − x
2
− y
2
1/2
with 1 − x
2
− z
2 > 0 and 1 − x
2
− y
2 > 0.
This leads to a coherent definition of functions of class C
r on a random
open subset of the sphere.
It can also be formulated more directly. First, given a C
r function of
(x, y, z) on an open subset V of R
3 , it restriction to U = V ∩ X is of class C
r
in the previous sense. Conversely, consider a C
r function f on an open subset
U of X and take a neighbourhood of a point (a, b, c) ∈ U . If, for example
c > 0, the definition shows that in the neighbourhood of (a, b, c), f is a C
r
function of (x, y) defined in the neighbourhood of a point (a, b) of R
2 . The
composition of f and the projection (x, y, z) → (x, y) of R
3 onto R
2 is a C
r
function of (x, y, z) on a vertical cylinder having an open subset of the plane
(x, y) as base. The restriction of this function to X is equal to the given
function f on a neighbourhood of (a, b, c). In conclusion, a fuction f defined
on an open subset U of X is of class C
r if and only if, in the neighbourhood
of every point of U , it has a C
r extension on an open subset of R
3 .
(ii) The notion of a manifold of class C
r and dimension d is obtained by
generalizing the construction of C
r functions on the sphere.
To start with, a manifold X is a separated topological space. Hence there
is a category of sets in X called open and satisfying the two obvious conditions
(any union of open sets is open, the intersection of a finite number of open
sets is open), as well as Hausdorff’s condition: if a and b are two distinct
points of X, there are open disjoint subsets U and V containing a and b.
Topological spaces are the natural realm of the notion of continuity: a map
f : X −→ Y is continuous if and only if the inverse image f
−1 (V ) of any
open subset V of Y is an open subset of X.
A differential manifold X must also be a locally Cartesian topological
space: for each a ∈ X, there must be an open neighbourhood U homeomor-