§ 4. Differential Manifolds
225
space R
d is a manifold if C
r functions are defined on it. Having said that, U
is an open Cartesian subset of a manifold X if and only if, as a manifold, U
is diffeomorphic to an open subset of R
d ; if, moreover, (U, ϕ) is a chart, then
ϕ is a diffeomorphism from U onto the open subspace ϕ(U ) and conversely.
Proofs reduce to rewording exercises.
Many other such trivialities can be found in detailed presentations, in
particular in volume 3 of Dieudonn´ e’s El´ ements d’analyse which, fortunately,
gives frequent illustrations in the form of examples or exercises that are
a great deal harder than the soporific but crucial definitions, scholia and
sorites
51 that we end up learning through repeated use.
In particular, it is possible to define the notion of a product manifold : take
two manifolds X and Y of class C
r , and as (U, ϕ) and (V, ψ) are charts of
X and Y with values in R
p and R
q , consider the map (x, y) → (ϕ(x), ψ(y))
from U × V to R
p+q . Making these charts vary gives an atlas of class C
r
for X × Y , whence a manifold structure on the topological space
52 X × Y .
You will have no difficulty in showing that a map z → (f (z), g(z)) from a
manifold Z to X × Y is of class C
r if and only if so are f : Z −→ X and
g : Z −→ Y , or that the projections X × Y −→ X are X × Y −→ Y are
of class C
r . And many more wonders. . . One may laugh, but this is what
transformed the loose theory available in the 1930s to a perfectly clear and
precise mechanism, whose concepts often suffice to indicate the notions that
should be introduced and the theorems that should be proved, at least at an
elementary level: they only need to well-defined.
12 – Tangent vectors and Differentials
(i) Vectors and tangent vector spaces. In what way can the calculations of the
preceding §§ be generalized to a d-dimensional manifold X, for example the
notion of a differential form ? We instantaneously encounter a fundamental
difficulty: it is possible to talk about “ vectors ” and “ linear forms ” in a
Cartesian space, but there are no vectors in a “ curved ” space, not even in a
sphere in R
3 . To bypass this obstacle, associate to each a ∈ X an “ abstract ”
vector space having the same dimension d as X, called the tangent vector
space to X at a and which I will denote X
(a), other authors adopting other
conventions, for example T a (X) which I will sometimes use.
51 Scholium : In philology, a grammatical or critical note explaining classical texts.
In geom. A remark on several propositions made in order to show their link, restriction or extension. Sorite : Sort of argument, in which a series of propositions
is so arranged that the second one must explain the predicate of the first one,
the third one the predicate of the second one, and so one, until the conclusion
wanted is reached. Predicate : In log. and gram. What is denied or affirmed of
the subject of a proposition. In the proposition : All men are mortals, mortals is
the predicate. (Littr´ e).
52 If X and Y are two topological spaces, ordaining a subset of X × Y to be open
if and only if it is a union of sets U × V , where U and V are open in X and Y
defines a topology on X × Y .
225
space R
d is a manifold if C
r functions are defined on it. Having said that, U
is an open Cartesian subset of a manifold X if and only if, as a manifold, U
is diffeomorphic to an open subset of R
d ; if, moreover, (U, ϕ) is a chart, then
ϕ is a diffeomorphism from U onto the open subspace ϕ(U ) and conversely.
Proofs reduce to rewording exercises.
Many other such trivialities can be found in detailed presentations, in
particular in volume 3 of Dieudonn´ e’s El´ ements d’analyse which, fortunately,
gives frequent illustrations in the form of examples or exercises that are
a great deal harder than the soporific but crucial definitions, scholia and
sorites
51 that we end up learning through repeated use.
In particular, it is possible to define the notion of a product manifold : take
two manifolds X and Y of class C
r , and as (U, ϕ) and (V, ψ) are charts of
X and Y with values in R
p and R
q , consider the map (x, y) → (ϕ(x), ψ(y))
from U × V to R
p+q . Making these charts vary gives an atlas of class C
r
for X × Y , whence a manifold structure on the topological space
52 X × Y .
You will have no difficulty in showing that a map z → (f (z), g(z)) from a
manifold Z to X × Y is of class C
r if and only if so are f : Z −→ X and
g : Z −→ Y , or that the projections X × Y −→ X are X × Y −→ Y are
of class C
r . And many more wonders. . . One may laugh, but this is what
transformed the loose theory available in the 1930s to a perfectly clear and
precise mechanism, whose concepts often suffice to indicate the notions that
should be introduced and the theorems that should be proved, at least at an
elementary level: they only need to well-defined.
12 – Tangent vectors and Differentials
(i) Vectors and tangent vector spaces. In what way can the calculations of the
preceding §§ be generalized to a d-dimensional manifold X, for example the
notion of a differential form ? We instantaneously encounter a fundamental
difficulty: it is possible to talk about “ vectors ” and “ linear forms ” in a
Cartesian space, but there are no vectors in a “ curved ” space, not even in a
sphere in R
3 . To bypass this obstacle, associate to each a ∈ X an “ abstract ”
vector space having the same dimension d as X, called the tangent vector
space to X at a and which I will denote X
(a), other authors adopting other
conventions, for example T a (X) which I will sometimes use.
51 Scholium : In philology, a grammatical or critical note explaining classical texts.
In geom. A remark on several propositions made in order to show their link, restriction or extension. Sorite : Sort of argument, in which a series of propositions
is so arranged that the second one must explain the predicate of the first one,
the third one the predicate of the second one, and so one, until the conclusion
wanted is reached. Predicate : In log. and gram. What is denied or affirmed of
the subject of a proposition. In the proposition : All men are mortals, mortals is
the predicate. (Littr´ e).
52 If X and Y are two topological spaces, ordaining a subset of X × Y to be open
if and only if it is a union of sets U × V , where U and V are open in X and Y
defines a topology on X × Y .