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⇔
∂U
L
=
e
LM
1
M
2
det [G
l
]
G
M
1
N
1
G
M
2
N
2
∂x
∂y
∂U
N
1
∂z
∂U
N
2
,
d
x
=
∗
(d
y ∧
d
z
)
.
Lemma 2.5 (
Zund
(
1987
),
Grafarend and Syffus
(
1998d
, p. 292), the Zund equations of a three-
dimensional conformeomorphism
3
l
r
=
3
,δ
μν
3
).
M
→
M
{
R
}
=
E
Equivalent formulations of the equations producing a conformal mapping
M
l
→
M
r
=
E
3
are
provided by the following formulations.
Formulation (i):
d
x
=
∗
(d
y
∧
d
z
)
.
(2.39)
Formulation (ii):
2
e
IJ
1
J
2
|
G
l
|G
J
1
K
1
G
J
2
K
2
∂x
∂y
∂U
k
1
∂z
∂U
K
2
.
∂U
I
=
1
∀I, J
1
,J
2
,K
1
,K
2
∈{
1
,
2
,
3
}
:
(2.40)
Formulation (iii):
(
G
21
G
32
G
31
G
12
)
∂y
∂U
G
31
G
23
)
∂y
∂U
∂V
+(
G
21
G
33
∂z
∂z
−
−
∂W
+
2
∂x
∂U
=
1
G
32
G
21
)
∂y
∂V
G
32
G
23
)
∂y
∂V
+(
G
22
G
31
∂U
+(
G
22
G
33
∂z
∂z
|
G
l
|
−
−
∂W
+
(2.41)
∂V
,
G
33
G
21
)
∂y
∂W
G
33
G
22
)
∂y
∂W
+(
G
23
G
31
∂U
+(
G
23
G
32
∂z
∂z
−
−
(
G
31
G
12
G
11
G
32
)
∂y
∂U
G
11
G
33
)
∂y
∂U
∂V
+(
G
31
G
13
∂z
∂z
−
−
∂W
+
2
|
G
l
|
∂x
∂V
=
1
G
12
G
31
)
∂y
∂V
G
12
G
33
)
∂y
∂V
+(
G
32
G
11
∂U
+(
G
32
G
13
∂z
∂z
−
−
∂W
+
(2.42)
∂V
,
G
13
G
31
)
∂y
∂W
G
13
G
32
)
∂y
∂W
+(
G
33
G
11
∂U
+(
G
33
G
12
∂z
∂z
−
−
(
G
11
G
22
G
21
G
12
)
∂y
G
21
G
13
)
∂y
∂V
+(
G
11
G
23
∂z
∂z
−
−
∂W
+
∂U
∂U
2
∂W
=
1
∂x
G
22
G
11
)
∂y
G
22
G
13
)
∂y
|
G
l
|
+(
G
12
G
21
∂U
+(
G
12
G
23
∂z
∂z
(2.43)
−
−
∂W
+
∂V
∂V
∂V
,
G
23
G
11
)
∂y
G
23
G
12
)
∂y
+(
G
13
G
21
∂U
+(
G
13
G
22
∂z
∂z
−
−
∂W
∂W
subject to
1
|
G
l
|
1
|
G
l
|
G
11
=
(
G
22
G
33
− G
23
G
32
)
, G
12
=
(
G
13
G
32
− G
12
G
33
)
,
1
|
G
l
|
1
|
G
l
|
G
13
=
(
G
12
G
23
− G
13
G
22
)
, G
22
=
(
G
11
G
33
− G
13
G
31
)
,
(2.44)
1
|
G
l
|
1
|
G
l
|
G
23
=
(
G
12
G
31
− G
11
G
32
)
, G
33
=
(
G
11
G
22
− G
12
G
21
)
.
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