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g
11
p
v
+
g
22
p
u
−
g
12
p
u
−
g
12
p
v
g
11
g
22
g
11
g
22
⇒
=0
.
(1.190)
g
12
g
12
−
−
v
u
End of Proof (the second step).
Proof (sketch of the proof for the third step).
λ
2
I
2
), namely
a conformal transformation from left conformal (isometric, isothermal) coordinates
2
Λ
2
I
2
)
The special KL equations generate a conformal mapping
M
l
(
P,Q
|
→
M
r
(
p, q
|
{
P,Q
}
to
right conformal (isometric, isothermal) coordinates
. The left matrix of the conformally flat
metric,
Λ
2
I
2
, is transformed to the right matrix of the conformally flat metric,
λ
2
I
2
.Uptothe
factors of conformality,
Λ
2
(
P,Q
)and
λ
2
(
p, q
), the matrices of the left and right matrices are unit
matrices, I
2
. Here, we only outline how the
integrability conditions p
PQ
=
p
QP
and
q
PQ
=
q
QP
are
converted to the special Laplace-Beltrami equation.
{
p, q
}
KL, 1st equation and 2nd equation, lead to the following relations.
1st :
p
PQ
=
p
QP
,p
PQ
=
q
QQ
,p
QP
=
−q
PP
,
(1.191)
p
PQ
=
p
QP
⇔ q
QQ
=
−q
PP
⇒ q
PP
+
q
QQ
=0
.
2nd :
q
PQ
=
q
QP
,q
QP
=
p
PP
,−q
PQ
=
p
QQ
,
(1.192)
q
QP
=
q
PQ
⇔
p
PP
+
p
QQ
=0
.
This concludes the proofs.
p
PP
=
−
p
QQ
⇒
End of Proof (the second step).
Note that a more elegant proof of the Korn-Lichtenstein equations based upon
exterior calcu-
lus
has been presented by
Grafarend and Syffus
(
1998d
). In addition, the authors succeeded to
generalize the fundamental differential equations which govern a conformeomorphism the num-
ber of dimensions being
n
(for
n
= 3, they coincide with the
Zund equations
(
Zund 1987
)from
M
3
3
n
l
l
to
M
r
), namely left (pseudo-)Riemann manifold
M
→
right (pseudo-)Riemann manifold
r,s
(
r
+
s
=
n
). In general, conformal mappings from an arbitrary left (pseudo-)Riemann
manifold
r
:=
M
E
r
do not exist. The dimension
n
= 2 is just an exception where conformal mappings always exist, though may be dicult to find.
For instance, due to involved diculties. The Philosophical Faculty of the University of Goettingen
Georgia Augusta (dated 13 June 1857) set up the “Preisaufgabe” to find a conformal mapping of
the triaxial ellipsoid which had already parameterized by C.F. Gauss in terins of “surface normal
coordinates” applying the “Gauss map”. Based upon the Jacobi's contribution on elliptic coordi-
nates (
Jacobi 1839
), which separate the Laplace-Beltrami equations of harmonicity, the “Preiss-
chrift” of
Schering
(
1857
) was finally crowned, nevertheless leaving the numerical problem open as
to how to construct a conformal map of the triaxial ellipsoid—up to now an open problem (
Klin-
genberg 1982
;
Schmehl 1927
;
Mueller 1991
).Thecaseofdimension
n
= 3 is a special case to be
treated. In contrast, for dimension
n>
3, a general statement can be made: a conformeomorphism
exists
if and only if
the
Weyl curvature tensor
, being a curvature element of the
Riemann curva-
ture tensor
, vanishes. We have given in Table
1.4
a list of related, commented references. A typical
example for the non-existence of a conformeomorphism is provided by the following example.
n
M
l
to an arbitrary right (pseudo-)Riemann manifold
M
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