68
1 From Riemann manifolds to Riemann manifolds
Box 1.35 (Representation of the factors of conformality in terms of conformal coordinates).
Left factor of conformality:
P (Λ, Φ) = f (Φ) cos Λ , Q(Λ, Φ) = f (Φ) sin Λ ,
λ
2
l =
A
2
1 cos
2 Φ
1 − E 2 sin
2 Φ
1
f 2 (Φ)
=
A
2
1 cos
2 f
−1
“ p
P 2 + Q 2
”
1 − E 2 sin
2 f −1
“ p
P 2 + Q 2
”
1
P 2 + Q 2 ,
Λ
2
l =
1 − E
2 sin
2 f
−1
“ p
P 2 + Q 2
”
A
2
1 cos 2 f −1
“ p
P 2 + Q 2
”
`
P
2 + Q
2 ´
.
(1.251)
Right factor of conformality:
p(λ, φ) = 2r tan
„
π
4
−
φ
2
«
cos λ , q(λ, φ) = 2r tan
„
π
4
−
φ
2
«
sin λ , tan(α/2) =
r
1 − cos α
1 + cos α
⇔
tan
„
π
4
−
φ
2
«
= tan
1
2
“ π
2
− φ
”
=
r
1 − sin α
1 + sin α
, 2r tan
„
π
4
−
φ
2
«
= 2r
r
1 − sin α
1 + sin α
=
p
p 2 + q 2
⇒
sin φ =
4r
2 − (p
2 + q
2 )
4r 2 + (p 2 + q 2 )
, cos φ =
4r
p
p 2 + q 2
4r 2 + (p 2 + q 2 )
,
sin λ =
tan λ
√
1 + tan
2 λ
=
q
p
p 2 + q 2
, cos λ =
1
√
1 + tan
2 λ
=
p
p
p 2 + q 2
,
λ
2
r = cos
4 ` π
4
−
φ
2
´
=
1
4
`
1 + sin φ
´ 2 =
16r
4
(4r 2 + p 2 + q 2 )
2 ,
Λ
2
r =
1
cos 4
` π
4
−
φ
2
´ =
4
`
1 + sin φ
´ 2 =
`
4r
2 + p
2 + q
2
´ 2
16r 4
.
(1.252)
Box 1.36 (The differential equation which governs the factor of conformality).
Two versions of the special Helmholtz equations (k is the Gaussian curvature k(p, q)):
(i) ∆ ln λ
2 + 2kλ
2 = 0 . (ii) ∆λ
2 + 2kλ
4 = 0 .
(1.253)
Right differential equation of the factor of conformality (S
2
r ):
k r =
1
r 2 = constant , ∆ ln λ
2
r +
2
r 2 λ
2
r = 0 ,
λ
2
r =
16r
4
(4r 2 + p 2 + q 2 )
2 , ln λ
2
r = ln 16r
4 − 2 ln
`
4r
2 + p
2 + q
2 ´
,
D p ln λ
2
r = −4
p
4r 2 + p 2 + q 2 , D q ln λ
2
r = −4
q
4r 2 + p 2 + q 2 ,
D pp ln λ
2
r = D
2
p ln λ
2
r = −4
4r
2 + p
2 + q
2 − 2p
2
(4r 2 + p 2 + q 2 )
2
, D qq ln λ
2
r = D
2
q ln λ
2
r = −4
4r
2 + p
2 + q
2 − 2q
2
(4r 2 + p 2 + q 2 )
2
,
D pp ln λ
2
r = −
4
(4r 2 + p 2 + q 2 )
2
`
4r
2 − p
2 + q
2 ´
, D qq ln λ
2
r = −
4
(4r 2 + p 2 + q 2 )
2
`
4r
2 + p
2 − q
2 ´
,
∆ r ln λ
2
r = −
32r
2
(4r 2 + p 2 + q 2 )
2 = −
2
r 2 λ
2
r
q. e. d.
(1.254)