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In-Depth Information
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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