224
IX – Multivariate Differential and Integral Calculus
delight undergraduates, including the present author during his youth. We
thus learnt that all circles in the plane, irrespective of their centre, go through
the same two points at infinite, the “ cyclic points ” with homogeneous coordinates (1, i, 0) and (1, −i, 0). These manifolds play an important role in
algebraic geometry and their topological properties have been extensively
studied.
(iv) Differentiable maps. In the same way that topological spaces are
adapted to the general notion of a continuous map, manifolds are adapted
to the notion of a differentiable map. It is all based on the following remark:
given two manifolds X and Y and a map f : X −→ Y taking the domain
U of a chart (U, ϕ) to X in the domain V of a chart (V, ψ) of Y , there is a
unique map F : ϕ(U ) −→ ψ(V ) such that
ψ ◦ f = F ◦ ϕ
on U ; it is the map which, for any x ∈ U , makes it possible to calculate
the coordinates η
j of the point y = f (x) in the chart (V, ψ) in terms of the
coordinates ξ
i of x in the chart (U, ϕ). We will sometimes say that F expresses
f in the charts considered. Note that, if f is continuous at a point a of X,
then for all charts (V, ψ) of Y at b = f (a), there exists a chart (U, ϕ) of
X at a such that f (U ) ⊂ V , since f
−1 (V ) is a neighbourhood of a, and so
contains an open set containing a, which contains the domain of a chart of
X at a. Because of this trivial observation, it is possible to generalize the
definitions and results relating to Cartesian spaces to maps from a manifold
to another, provided we check they have are well-defined independently of
the charts used, a fact generally immediate.
For example, given two manifolds X and Y of class at least C
r , f will
be said to be of class C
r if f is continuous and if, for any charts (U, ϕ) and
(V, ψ) such that f (U ) ⊂ V , the function F expressing f in these charts is of
class C
r in the usual sense. This means that, for any open subset W of Y
and any function g ∈ C
r (W ), the composite function g ◦ f , defined on the
open subset f
−1 (W ), is of class C
r on it. If, moreover, f is a homeomorphism
and if f
−1 is of class C
r , f is said to be a diffeomorphism of class C
r from
X to Y . The reader will easily check that the map X −→ Z obtained by
composing two maps X −→ Y and Y −→ Z of class C
r is also of class C
r .
Maps of class C
r are also called homomorphisms of manifolds, in line
with homomorphisms of groups, rings or vector spaces, etc. in algebra. The
Grothendieck school censored the controversial prefix “ homo ” and invented
the term morphism, which the Greeks would have probably considered doubly
barbarian.
50
This definition makes the characterization of open Cartesian subsets of X
possible. First, any open subset U of X is itself a manifold since there are C
r
functions on any smaller open subset. In particular, any open subset U of a
50 Barbarian : a foreigner, with respect to the Greeks and Romans (Littr´ e).
IX – Multivariate Differential and Integral Calculus
delight undergraduates, including the present author during his youth. We
thus learnt that all circles in the plane, irrespective of their centre, go through
the same two points at infinite, the “ cyclic points ” with homogeneous coordinates (1, i, 0) and (1, −i, 0). These manifolds play an important role in
algebraic geometry and their topological properties have been extensively
studied.
(iv) Differentiable maps. In the same way that topological spaces are
adapted to the general notion of a continuous map, manifolds are adapted
to the notion of a differentiable map. It is all based on the following remark:
given two manifolds X and Y and a map f : X −→ Y taking the domain
U of a chart (U, ϕ) to X in the domain V of a chart (V, ψ) of Y , there is a
unique map F : ϕ(U ) −→ ψ(V ) such that
ψ ◦ f = F ◦ ϕ
on U ; it is the map which, for any x ∈ U , makes it possible to calculate
the coordinates η
j of the point y = f (x) in the chart (V, ψ) in terms of the
coordinates ξ
i of x in the chart (U, ϕ). We will sometimes say that F expresses
f in the charts considered. Note that, if f is continuous at a point a of X,
then for all charts (V, ψ) of Y at b = f (a), there exists a chart (U, ϕ) of
X at a such that f (U ) ⊂ V , since f
−1 (V ) is a neighbourhood of a, and so
contains an open set containing a, which contains the domain of a chart of
X at a. Because of this trivial observation, it is possible to generalize the
definitions and results relating to Cartesian spaces to maps from a manifold
to another, provided we check they have are well-defined independently of
the charts used, a fact generally immediate.
For example, given two manifolds X and Y of class at least C
r , f will
be said to be of class C
r if f is continuous and if, for any charts (U, ϕ) and
(V, ψ) such that f (U ) ⊂ V , the function F expressing f in these charts is of
class C
r in the usual sense. This means that, for any open subset W of Y
and any function g ∈ C
r (W ), the composite function g ◦ f , defined on the
open subset f
−1 (W ), is of class C
r on it. If, moreover, f is a homeomorphism
and if f
−1 is of class C
r , f is said to be a diffeomorphism of class C
r from
X to Y . The reader will easily check that the map X −→ Z obtained by
composing two maps X −→ Y and Y −→ Z of class C
r is also of class C
r .
Maps of class C
r are also called homomorphisms of manifolds, in line
with homomorphisms of groups, rings or vector spaces, etc. in algebra. The
Grothendieck school censored the controversial prefix “ homo ” and invented
the term morphism, which the Greeks would have probably considered doubly
barbarian.
50
This definition makes the characterization of open Cartesian subsets of X
possible. First, any open subset U of X is itself a manifold since there are C
r
functions on any smaller open subset. In particular, any open subset U of a
50 Barbarian : a foreigner, with respect to the Greeks and Romans (Littr´ e).