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In-Depth Information
(i) Special KL:
q u
q υ
,
g 11 g 22 −g 12
1
p u
p υ
=
p u =
(
g 12 q u + g 11 q υ )
g 12 g 11
.
g 11 g 22 −g 12
1
(1.173)
g 11 g 22 −g 12
1
g 22 g 12
p υ =
(
g 12 q u + g 12 q υ )
(ii)Right integrability :
p = p υu and q = q υu
or (in termsof theLaplace −− Beltrami operator)
(1.174)
Δ p := g 11 p υ −g 12 p u
+ g 22 p u −g 12 p υ
g 11 g 22 −g 12
g 11 g 22 −g 12
=0
Δ q := g 11 p υ −g 12 p u
+ g 22 q u −g 12 q υ
υ
u
.
(1.175)
g 11 g 22 −g 12
g 11 g 22 −g 12
=0
υ
u
(iii) Right orientation conservation:
p u p υ
q u q υ
= p u q υ − p υ q u > 0 .
(1.176)
3rd step (left-right).
The left Riemann manifold M
l ( P,Q|Λ 2 I 2 ) which here is called left surface and is parameter-
ized in left conformal coordinates {P,Q} , is orientation preserving conformally mapped onto the
right Riemann manifold M
| 2 ), which here is called right surface and is parameterized
in right conformal coordinates {p, q} , if the following special Korn-Lichtenstein equations (i)
(called Cauchy-Riemann (or d'Alembert-Euler) equations) subject to the following integrability
conditions of harmonicity (ii) and orientation conservation (iii) are solved.
r ( p, q|λ 2
(i) Special KL: (Cauchy-Riemann, d'Alembert-Euler):
p P
p Q
=
01
10
q P
q Q
,p P = q Q and p Q =
1
g 11 g 22
q P .
(1.177)
g 12
(ii) Right integrability:
p PQ = p QP and q PQ = q QP
(1.178)
or (in terms of the Laplace-Beltrami operator)
Δ PQ p := p PP + p QQ 2
p ( P,Q )=0
2
∂Q 2
∂P 2 +
Δ PQ q := q PP + q QQ 2
q ( P,Q )=0
.
(1.179)
2
∂Q 2
∂P 2 +
(iii) Left orientation conservation:
p P p Q
q P q Q
= p P q Q
p Q q P =0 .
(1.180)
The special Korn-Lichtenstein equations, which govern as Cauchy-Riemann (or d'Alembert-
Euler) equations any harmonic, orientation preserving conformal mapping M
l ( P,Q ) M
r ( p, q ),
are uniquely solvable if a proper boundary value problem is formulated.
End of Theorem.
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