60
1 From Riemann manifolds to Riemann manifolds
Box 1.28 (Representation of the factors of conformality in terms of conformal coordinates).
Left factor of conformality:
P = A 1 Λ , Q = A 1 f (Φ) , f(Φ) := ln tan
„
π
4
+
Φ
2
« »
1 − E sin Φ
1 + E sin Φ
– E/2
!
,
λ
2
l =
cos
2 Φ
1 − E 2 sin
2 Φ
, Λ
2
l =
1 − E
2 sin
2 Φ
cos 2 Φ
⇒
λ
2
l =
cos
2 f
−1 (Q/A 1 )
p
1 − E 2 sin
2 f −1 (Q/A 1 )
, Λ
2
l =
1 − E
2 sin
2 f
−1 (Q/A 1 )
cos 2 f −1 (Q/A 1 )
.
(1.217)
Right factor of conformality:
p = rλ , q = r ln tan
„
π
4
+
φ
2
«
= r artanh sin φ ,
tanh(q/r) = sin φ ,
1
cosh(q/r)
= cos φ ,
λ
2
r = cos
2 φ , Λ
2
r =
1
cos 2 φ
⇒
λ
2
r =
1
cosh
2 (q/r)
, Λ
2
r = cosh
2 (q/r) .
(1.218)
Box 1.29 (The differential equation which governs the factor of conformality).
Two versions of the special Helmholtz equations:
(i) ∆ ln λ
2 + 2kλ
2 = 0 , (ii) ∆λ
2 + 2kλ
4 = 0 .
(1.219)
(k is the Gaussian curvature k(p, q).)
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 = cosh
−2 (q/r) , ln λ
2
r = −2 ln cosh(q/r) ,
D q ln λ
2
r = −
2
r
tanh(q/r) , ∆ r ln λ
2
r = D qq ln λ
2
r = −
2
r 2
1
cosh
2 (q/r)
= −
2
r 2 λ
2
r
(1.220)
q. e. d.
Left differential equation of the factor of conformality (E
2
A 1 ,A 1 ,A 2 ):
k l =
(1 − E
2 sin
2 φ)
2
A
2
1 (1 − E 2 )
=
1 − E
2 sin
2 f
−1 (Q/A 1 )
A
2
1 (1 − E 2 )
,
∆ ln λ
2
l + 2k(Q)λ
2
l = 0 , λ
2
l =
cos
2 f
−1 (Q/A 1 )
p
1 − E 2 sin
2 f −1 (Q/A 1 )
,
ln λ
2
l = 2 ln f
−1 (Q/A 1 ) −
1
2
ln
ˆ
1 − E
2 sin
2 f
−1 (Q/A 1 )
˜
,
∆ l ln λ
2
l = D QQ ln λ
2
l = −2k(Q)λ
2
l
(1.221)
q. e. d.