§ 4. Differential Manifolds
249
are at least C
2 on each chart (U, ϕ). If h = h
i a i (ξ) and k = k
i a i (ξ) are two
tangent vectors to X at x, then
(h|k) = g ij (ξ)h
i k
j
and in particular, for the metaphysical vector dx = a i (ξ)dξ
i ,
ds
2 = (dx|dx) = g ij (ξ)dξ
i dξ
j ,
i.e. the square of the length of dx. The length of a curve μ : [a, b] → X can
then be defined by the formula
m(μ) =
b
a
(μ
(t)|μ
(t))
1/2 dt
and it is possible to look for the geodesics, i.e. the curves of minimal length
connecting two given points of X, if they exist. It is also possible to define the
covariant derivative of a tensor field, etc. All this has given rise to an extensive
literature whose latest avatar seems to be Serge Lang’s book, Riemannian
Geometry (Springer, 1999).
If X is a submanifold of an Euclidean space E (i.e. a Cartesian space
equipped with a Hilbert scalar product) and if h, k ∈ X
(x) are defined by
the curves μ and ν as mentioned in (iii) of n
◦ 12, it is natural to set
(h|k) = (μ
(0)|ν
(0)) ,
where μ
(0) and ν
(0) are defined as limits. Equivalently, note that X
(x) can
be canonically identified to a vector subspace of E
(x), hence to E, which
gives the scalar product sought in X
(x). The explicit computation of the ds
2
of X is particularly simple when X is defined by a parametric representation
x = σ(t) in the neighbourhood of point a, where t varies in an open subset of
R
d and where σ is an open immersion . Then X
(x) is the image of R
d under
σ
(t) if x = σ(t), so that if, in Leibniz style, we set dx = σ
(t)dt, then
ds
2 = (σ
(t)dt|σ
(t)dt) .
For example, for the unit sphere in R
3 in spherical coordinates
x = cos ϕ cos ψ , y = sin ϕ cos ψ , z = sin ψ ,
we differentiate x, y and z, we simply calculate dx
2 + dy
2 + dz
2 and we find
that ds
2 = cos
2 ψdϕ
2 + dψ
2 .
It can be shown that for any connected Riemann space X, there is a
diffeomorphism from X onto a submanifold Y of some space R
n which transforms the given ds
2 in X into that of Y (John Nash, 1956). Dieudonn´ e (vol. 4,
XX.15) proves a far weaker result ( ´
E. Cartan) as it is purely local, but it is
already difficult.
249
are at least C
2 on each chart (U, ϕ). If h = h
i a i (ξ) and k = k
i a i (ξ) are two
tangent vectors to X at x, then
(h|k) = g ij (ξ)h
i k
j
and in particular, for the metaphysical vector dx = a i (ξ)dξ
i ,
ds
2 = (dx|dx) = g ij (ξ)dξ
i dξ
j ,
i.e. the square of the length of dx. The length of a curve μ : [a, b] → X can
then be defined by the formula
m(μ) =
b
a
(μ
(t)|μ
(t))
1/2 dt
and it is possible to look for the geodesics, i.e. the curves of minimal length
connecting two given points of X, if they exist. It is also possible to define the
covariant derivative of a tensor field, etc. All this has given rise to an extensive
literature whose latest avatar seems to be Serge Lang’s book, Riemannian
Geometry (Springer, 1999).
If X is a submanifold of an Euclidean space E (i.e. a Cartesian space
equipped with a Hilbert scalar product) and if h, k ∈ X
(x) are defined by
the curves μ and ν as mentioned in (iii) of n
◦ 12, it is natural to set
(h|k) = (μ
(0)|ν
(0)) ,
where μ
(0) and ν
(0) are defined as limits. Equivalently, note that X
(x) can
be canonically identified to a vector subspace of E
(x), hence to E, which
gives the scalar product sought in X
(x). The explicit computation of the ds
2
of X is particularly simple when X is defined by a parametric representation
x = σ(t) in the neighbourhood of point a, where t varies in an open subset of
R
d and where σ is an open immersion . Then X
(x) is the image of R
d under
σ
(t) if x = σ(t), so that if, in Leibniz style, we set dx = σ
(t)dt, then
ds
2 = (σ
(t)dt|σ
(t)dt) .
For example, for the unit sphere in R
3 in spherical coordinates
x = cos ϕ cos ψ , y = sin ϕ cos ψ , z = sin ψ ,
we differentiate x, y and z, we simply calculate dx
2 + dy
2 + dz
2 and we find
that ds
2 = cos
2 ψdϕ
2 + dψ
2 .
It can be shown that for any connected Riemann space X, there is a
diffeomorphism from X onto a submanifold Y of some space R
n which transforms the given ds
2 in X into that of Y (John Nash, 1956). Dieudonn´ e (vol. 4,
XX.15) proves a far weaker result ( ´
E. Cartan) as it is purely local, but it is
already difficult.