2-3 The equivalence theorem for conformal mappings 105
Lemma 2.5 (J. Zund (1987), E. Grafarend and R. Syffus (1998d, p. 292), the Zund equations of a threedimensional conformeomorphism M
3
l → M
3
r = {R
3 , δ µν } = E
3 ).
Equivalent formulations of the equations producing a conformal mapping M
3
l → M
3
r = E
3 are provided
by the following formulations.
Formulation (i):
dx = ∗(dy ∧ dz) .
(2.39)
Formulation (ii):
∀ I, J 1 , J 2 , K 1 , K 2 ∈ {1, 2, 3} :
∂x
∂U I =
1
2 e IJ 1 J 2
|G l |G
J 1 K 1 G
J 2 K 2 ∂y
∂U K 1
∂z
∂U K 2 .
(2.40)
Formulation (iii):
∂x
∂U =
1
2
|G l |
G
21 G
32
− G
31 G
12
∂y
∂U
∂z
∂V +
G
21 G
33
− G
31 G
23
∂y
∂U
∂z
∂W +
+
G
22 G
31
− G
32 G
21
∂y
∂V
∂z
∂U +
G
22 G
33
− G
32 G
23
∂y
∂V
∂z
∂W +
+
G
23 G
31
− G
33 G
21
∂y
∂W
∂z
∂U +
G
23 G
32
− G
33 G
22
∂y
∂W
∂z
∂V
,
(2.41)
∂x
∂V =
1
2
|G l |
G
31 G
12
− G
11 G
32
∂y
∂U
∂z
∂V +
G
31 G
13
− G
11 G
33
∂y
∂U
∂z
∂W +
+
G
32 G
11
− G
12 G
31
∂y
∂V
∂z
∂U +
G
32 G
13
− G
12 G
33
∂y
∂V
∂z
∂W +
+
G
33 G
11
− G
13 G
31
∂y
∂W
∂z
∂U +
G
33 G
12
− G
13 G
32
∂y
∂W
∂z
∂V
,
(2.42)
∂x
∂W =
1
2
|G l |
G
11 G
22
− G
21 G
12
∂y
∂U
∂z
∂V +
G
11 G
23
− G
21 G
13
∂y
∂U
∂z
∂W +
+
G
12 G
21
− G
22 G
11
∂y
∂V
∂z
∂U +
G
12 G
23
− G
22 G
13
∂y
∂V
∂z
∂W +
+
G
13 G
21
− G
23 G
11
∂y
∂W
∂z
∂U +
G
13 G
22
− G
23 G
12
∂y
∂W
∂z
∂V
,
(2.43)
subject to
G
11 =
1
|G l | (G 22 G 33 − G 23 G 32 ) , G
12 =
1
|G l | (G 13 G 32 − G 12 G 33 ) ,
G
13 =
1
|G l | (G 12 G 23 − G 13 G 22 ) , G
22 =
1
|G l | (G 11 G 33 − G 13 G 31 ) ,
G
23 =
1
|G l | (G 12 G 31 − G 11 G 32 ) , G
33 =
1
|G l | (G 11 G 22 − G 12 G 21 ) .
(2.44)
Formulation (iv):
∂x
∂U =
1
√
|G l |
G 11
∂y
∂V
∂z
∂W −
∂y
∂W
∂z
∂V
+ G 12
∂y
∂W
∂z
∂U −
∂y
∂U
∂z
∂W
+ G 13
∂y
∂U
∂z
∂V −
∂y
∂V
∂z
∂U
,
∂x
∂V =
1
√
|G l |
G 12
∂y
∂V
∂z
∂W −
∂y
∂W
∂z
∂V
+ G 22
∂y
∂W
∂z
∂U −
∂y
∂U
∂z
∂W
+ G 23
∂y
∂U
∂z
∂V −
∂y
∂V
∂z
∂U
,
∂x
∂W =
1
√
|G l |
G 13
∂y
∂V
∂z
∂W −
∂y
∂W
∂z
∂V
+ G 23
∂y
∂W
∂z
∂U −
∂y
∂U
∂z
∂W
+ G 33
∂y
∂U
∂z
∂V −
∂y
∂V
∂z
∂U
,
(2.45)
subject to the integrability conditions
∂
2 x
∂U∂V =
∂
2 x
∂V ∂U ,
∂
2 x
∂U∂W =
∂
2 x
∂W ∂U ,
∂
2 x
∂V ∂W =
∂
2 x
∂W ∂V .
End of Lemma.