222
IX – Multivariate Differential and Integral Calculus
with x
2 + y
2 + z
2 = 1. But considering any of the six local charts defined
above, the corresponding coordinates are two of the variables x, y, z, the third
one being, as we have seen, a C
∞ function of the other two. The chart ϕ is,
therefore, C
∞ - compatible with the atlas initially used since z = 1 outside
the north pole and conversely. Hence the differentiable structure of S could
have been defined by using these two stereographic projections.
To find an example of an “ abstract ” manifold, i.e. which is not given as
a subset of a Cartesian space, consider the projective space X = P n (R) ; by
definition, it is the set of 1-dimensional vector subspaces (lines with initial
point the origin) of R
n+1 . It would amount to the same to establish the
equivalence relation “ x and y are proportional ” on the set of non-zero vectors
of R
n+1 and to say that P n (R) is the quotient space of R
n+1
− {0} modulo
this relation would come to the same. An element of P n (R) is, therefore,
characterized by n + 1 numbers x
0 , . . . , x
n , not all of which are zero, defined
up to a factor; these are its homogeneous coordinates . Any function f defined
on a subset of P n (R) can be identified with a homogeneous function of these
coordinates, i.e. such that
f
tx
0 , . . . , tx
n
= f
x
0 , . . . , x
n
for all t = 0 .
If p(x) denotes the image of x ∈ R
n+1
− {0} in X = P n (R), i.e. the
subspace D generated by x, then a topology can be defined on X by requiring
U ⊂ X to be open if and only p
−1 (U ) is open in R
n+1
− {0} or, equivalently,
if the union of the lines D ∈ U is an open subset of R
n+1
− {0}. In particular,
X is the union of the n + 1 open sets U i (0 ≤ i ≤ n) that are images under p
of the open subsets of R
n+1 defined by x
i
= 0. As p(x) = p(x/x
i ), U i = p(E i )
also holds, where E i is the hyperplane with equation ξ
i = 1 . U i is, therefore,
the set of lines D having non-trivial intersection with E i and as such a line is
determined by its (unique) intersection point with E i , the map p : E i −→ U i
is bijective . This leads to an inverse map q i : U i −→ E i which takes every
line D ∈ U i onto its intersection point with E i . For all D ∈ U i ,
q i (D) =
ξ
0 , . . . , 1, . . . , ξ
n
with well-defined ξ
p = x
p /x
i , so that the formula
ϕ i (D) =
ξ
0 , . . . , ξ
i−1 , ξ
i+1 , . . . , ξ
n
gives a bijection from U i onto R
n , which together with its inverse is obviously
continuous. Hence the couples (U i , ϕ i ) thus defined are charts of X of class
C
0 covering P n (R). In fact, they are pairwise C
∞ -compatibles. Indeed, for
D ∈ U i ∩ U j , there are relations of the form
q i (D) =
ξ
0 , . . . , ξ
i−1 , 1, ξ
i+1 , . . . , ξ
n
,
q j (D) = (η
0 , . . . , η
j−1 , 1, η
j+1 , . . . , η
n ) ;
these points of R
n+1 being on D, ξ
k = η
k /η
i and η
k = ξ
k /ξ
j for all k .
As U i ∩ U j corresponds to y ∈ E j such that η
i
= 0, the coordinates ϕ
i (D)
IX – Multivariate Differential and Integral Calculus
with x
2 + y
2 + z
2 = 1. But considering any of the six local charts defined
above, the corresponding coordinates are two of the variables x, y, z, the third
one being, as we have seen, a C
∞ function of the other two. The chart ϕ is,
therefore, C
∞ - compatible with the atlas initially used since z = 1 outside
the north pole and conversely. Hence the differentiable structure of S could
have been defined by using these two stereographic projections.
To find an example of an “ abstract ” manifold, i.e. which is not given as
a subset of a Cartesian space, consider the projective space X = P n (R) ; by
definition, it is the set of 1-dimensional vector subspaces (lines with initial
point the origin) of R
n+1 . It would amount to the same to establish the
equivalence relation “ x and y are proportional ” on the set of non-zero vectors
of R
n+1 and to say that P n (R) is the quotient space of R
n+1
− {0} modulo
this relation would come to the same. An element of P n (R) is, therefore,
characterized by n + 1 numbers x
0 , . . . , x
n , not all of which are zero, defined
up to a factor; these are its homogeneous coordinates . Any function f defined
on a subset of P n (R) can be identified with a homogeneous function of these
coordinates, i.e. such that
f
tx
0 , . . . , tx
n
= f
x
0 , . . . , x
n
for all t = 0 .
If p(x) denotes the image of x ∈ R
n+1
− {0} in X = P n (R), i.e. the
subspace D generated by x, then a topology can be defined on X by requiring
U ⊂ X to be open if and only p
−1 (U ) is open in R
n+1
− {0} or, equivalently,
if the union of the lines D ∈ U is an open subset of R
n+1
− {0}. In particular,
X is the union of the n + 1 open sets U i (0 ≤ i ≤ n) that are images under p
of the open subsets of R
n+1 defined by x
i
= 0. As p(x) = p(x/x
i ), U i = p(E i )
also holds, where E i is the hyperplane with equation ξ
i = 1 . U i is, therefore,
the set of lines D having non-trivial intersection with E i and as such a line is
determined by its (unique) intersection point with E i , the map p : E i −→ U i
is bijective . This leads to an inverse map q i : U i −→ E i which takes every
line D ∈ U i onto its intersection point with E i . For all D ∈ U i ,
q i (D) =
ξ
0 , . . . , 1, . . . , ξ
n
with well-defined ξ
p = x
p /x
i , so that the formula
ϕ i (D) =
ξ
0 , . . . , ξ
i−1 , ξ
i+1 , . . . , ξ
n
gives a bijection from U i onto R
n , which together with its inverse is obviously
continuous. Hence the couples (U i , ϕ i ) thus defined are charts of X of class
C
0 covering P n (R). In fact, they are pairwise C
∞ -compatibles. Indeed, for
D ∈ U i ∩ U j , there are relations of the form
q i (D) =
ξ
0 , . . . , ξ
i−1 , 1, ξ
i+1 , . . . , ξ
n
,
q j (D) = (η
0 , . . . , η
j−1 , 1, η
j+1 , . . . , η
n ) ;
these points of R
n+1 being on D, ξ
k = η
k /η
i and η
k = ξ
k /ξ
j for all k .
As U i ∩ U j corresponds to y ∈ E j such that η
i
= 0, the coordinates ϕ
i (D)