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G 11 P V
,Q VU = G 22 P U
G 12 P U
G 12 P V
G 11 G 22
G 11 G 22
Q UV =
.
(1.184)
G 12
G 12
V
U
G 11 P V
= G 22 P U
G 12 P U
G 12 P V
2nd : Q UV = Q VU ⇔−
G 11 G 22
G 11 G 22
U
G 12
G 12
V
G 11 P V − G 12 P U
+ G 22 P U − G 12 P V
G 11 G 22 − G 12
G 11 G 22 − G 12
=0 .
(1.185)
V
U
End of Proof (the first step).
Proof (sketch of the proof for the second step).
λ 2I 2 ), a confor-
mal coordinate transformation from general right coordinates {u,v} to right conformal coordinates
{
The special KL equations generate the conformal mapping
M
r ( u,v
|
G r )
M
r ( p, q
|
. The right matrix of the metric, G r , is transformed to the right matrix of the conformally
flat metric , λ 2 I 2 .Uptothe factor of conformality , λ 2 ( p, q ), the transformed matrix of the metric
is a unit matrix, I 2 . Here, we only outline how the integrability conditions p uv = p vu and q uv = q vu
are converted to the Laplace-Beltrami equation.
p, q
}
KL, 1st equation and 2nd equation, lead to
p uv =
, vu =
g 12 q u + g 11 q v
g 22 q u + g 12 q v
g 11 g 22 − g 12
g 11 g 22 − g 12
.
(1.186)
v
u
g 11 q v
g 12 q u
g 22 q u
g 12 q v
g 11 g 22
g 11 g 22
1st : p uv = p vu
=
U
(1.187)
g 12
g 12
v
g 11 q v
+ g 2 2 q u
g 12 q u
g 11 g 22
g 12 q v
g 11 g 22
=0 .
g 12
g 12
v
u
The KL matrix equation is inverted to
q u
q v
=
g 12 −g 11
g 22 −g 12
p u
p v
, u = g 12 p u
1
g 11 p v
g 11 g 22
g 11 g 22
,
g 12
g 12
q v = g 22 p u
g 12 p v
g 11 g 22 − g 12
.
(1.188)
The inverted KL equations lead to
g 11 p v
, vu = g 22 p u
g 12 p u
g 12 p v
g 11 g 22
g 11 g 22
q uv =
.
(1.189)
g 12
g 12
g 11 p v
v
= g 2 2 p u
u
g 12 p u
g 11 g 22
g 12 p v
g 11 g 22
2nd : q uv = q vu ⇔−
u
g 12
g 12
v
 
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