106 2 From Riemann manifolds to Euclidean manifolds
Question.
Question: “Why did we bother you with the three-dimensional conformal mapping of a threedimensional Riemann manifold to a three-dimensional Euclidean manifold?” Answer: “One
of the main reasons is the inability of the theory of complex manifolds to work conformally
with odd-dimensional real manifolds. Only even-dimensional real manifolds M
2n (R) can be
transformed to complex manifolds M
n (C).”
Finally, Lemma 2.6 presents the partial differential equations of a conformeomorphism if it exists
from a left n-dimensional (pseudo-)Riemann manifold M
n
l of signature l to a right n-dimensional
(pseudo-)Riemann manifold M
n
r = E
n of signature r.
Lemma 2.6 (E. Grafarend and R. Syffus (1998d, p. 293), conformeomorphism).
Equivalent formulations of the equations producing a conformal mapping M
n
l → M
n
r = E
n are provided
by the following formulations.
Formulation (i):
dx
1 = ∗(dx
2
∧ . . . ∧ dx
n ) .
(2.46)
Formulation (ii):
∀ L, M 1 , . . . , M p , N 1 , . . . , N p ∈ {1, . . . , n}
(p = n − 1) :
∂x
∂U L =
1
p!
e LM 1 ...M p
det [G l ]G
M 1 N 1 . . . G
M p N p
∂x
2
∂U N 1
. . .
∂x
n
∂U N p ,
(2.47)
subject to the integrability conditions
∂
2 x
1
∂U L ∂U N =
∂
2 x
1
∂U N ∂U L .
(2.48)
End of Lemma.
2-4 The equivalence theorem for equiareal mappings
The equivalence theorem for equiareal mappings from the left two-dimensional Riemann manifold to
the right two-dimensional Euclidean manifold (areomorphism), Mollweide projection of the ellipsoid-ofrevolution, principal stretches.
The previous equivalence theorem for an areomorphism is specialized for the case of the twodimensional right Euclidean manifold {M
2
r , g µν } = {R
2 , δ µν } =: E
2 . In many applications, the choice
of {R
2 , δ µν } is the planar manifold, for instance, the tangent space T U 0 M
2
l of the left two-dimensional
Riemann manifold fixed to the point U 0 = {U
1
0 , U
2
0 }, being covered by Cartesian or polar coordinates.
For an illustration of such a setup of a “planar manifold”, go back to our previous examples. Here,
we focus on the equivalence theorem, namely the differential equations which govern an equiareal
mapping M
2
l → {R
2 , δ µν }.