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Formulation (iv):
G
11
∂y
+
G
12
∂y
∂x
∂U
=
1
|
∂y
∂W
−
∂y
∂W
∂y
∂V
∂z
∂U
−
∂y
∂U
∂z
∂W
∂V
∂W
G
l
|
+
G
13
∂y
,
∂z
∂V
−
∂y
∂V
∂z
∂U
∂U
G
12
∂y
∂V
+
G
22
∂y
∂W
∂x
∂V
=
1
∂z
∂W
−
∂y
∂W
∂z
∂V
∂z
∂U
−
∂y
∂U
∂z
∂W
|
G
l
|
+
G
23
∂y
∂U
,
∂z
∂V
−
∂y
∂V
∂z
∂U
(2.45)
G
13
∂y
+
G
23
∂y
∂x
∂W
=
1
|
∂z
∂W
−
∂y
∂W
∂z
∂V
∂z
∂U
−
∂y
∂U
∂z
∂W
∂V
∂W
G
l
|
+
G
33
∂y
,
∂z
∂V
−
∂y
∂V
∂z
∂U
∂U
∂
2
x
∂
2
x
∂
2
x
∂x
2
∂
2
x
∂
2
x
subject to the integrability conditions
∂U∂V
=
∂V ∂U
,
∂U∂W
=
∂W∂U
,
∂V ∂W
=
∂W∂V
.
End of Lemma.
Question: “Why did we bother you with the three-
dimensional conformal mapping of a three-dimensional
Riemann manifold to a three-dimensional Euclidean mani-
fold?” 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
2
n
(
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
l
of signature
l
toaright
n
-dimensional
(pseudo-)Riemann manifold
M
r
=E
n
of signature r.
Lemma 2.6 (
Grafarend and Syffus
(
1998d
, p. 293), conformeomorphism).
Equivalent formulations of the equations producing a conformal mapping
M
l
→
M
r
=
E
n
are
provided by the following formulations.
Formulation (i):
d
x
1
=
∗
(d
x
2
∧ ...∧
d
x
n
)
.
(2.46)
Formulation (ii):
∀
L, M
1
, ...., M
p
, N
1
, ..., N
p
∈{
1
, ..., n
}
(
p
=
n
−
1) :
(2.47)
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
L
=
1
∂x
∂U
N
p
,
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