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r
rests upon the
existence theorem of
Chern
(
1955a
,
b
), where it is shown that under rather mild certainty
assumptions, namely
C
2
,x
, conformal coordinates (isometric coordinates, isothermal coordi-
nates) exist as solutions of the left or right Korn-Lichtenstein equations. Let us here also
refer to the following authors.
Blaschke and Leichtweiß
(
1973
),
Boulware et al.
(
1970
),
Bour-
guignon
(
1970
),
Chen
(
1973
),
Chen and Yano
(
1973
),
Chern
(
1967a
),
Chern et al.
(
1954
),
Do
Carmo et al.
(
1985
),
Eisenhart
(
1949
),
Euler
(
1755
,
1777a
),
Ferrara et al.
(
1972
),
Finzi
(
1922
),
Gauss
(
1822
,
1844
),
Goenneretal.
(
1994
),
Jacobi
(
1839
),
Heitz
(
1988
),
Klingenberg
(
1982
),
Koenig
and Weise
(
1951a
),
Korn
(
1914
),
Krueger
(
1903
,
1922
),
Kuiper
(
1949
,
1950
),
Kulkarni
(
1969
,
1972
),
Kulkarni and Pinkall
(
1988
),
Lafontaine
(
1988a
,
b
),
de Lagrange
(
1781
),
Lancaster
(
1969
,
1973
),
Lichtenstein
(
1911
,
1916
),
Liouville
(
1850
),
Markuschewitsch
(
1955
),
Mirsky
(
1960
),
Mis-
ner
(
1978
),
Mitra and Rao
(
1968
),
De Moor and Zha
(
1991
),
Moore
(
1977
),
Nishikawa
(
1974
),
Ricci
(
1918
),
Riemann
(
1851
),
Samelson
(
1969
),
Schering
(
1857
),
Schmehl
(
1927
),
Schoen
(
1984
),
Schouten
(
1921
),
Spivak
(
1979
),
Stein and Weiss
(
1968
),
Weber
(
1867
),
Weyl
(
1918
,
1921
),
Wray
(
1974
),
Yano
(
1970
),
Yanushaushas
(
1982
),
Zadro and Carminelli
(
1966
), and
Zund
(
1987
).
The proof of the operational theorem of conformal mapping
M
l
→
M
Proof (sketch of the proof for the first step).
The special KL equations generate a conformal mapping,
M
l
(
U, V |
G
l
)
→
M
l
(
P,Q|Λ
2
I
2
), namely
a conformal coordinate transformation from general left coordinates
{U, V }
to left conformal coor-
dinates
{P,Q}
. The left matrix of the metric, i.e. the matrix G
l
, is transformed to the left 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.
KL, 1st equation and 2nd equation, lead to
P
UV
=
, P
VU
=
−
G
12
Q
U
+
G
11
Q
V
−
G
22
Q
U
+
G
12
Q
V
G
11
G
22
−
G
11
G
22
−
.
(1.181)
G
12
G
12
V
U
G
11
Q
V
−
G
12
Q
U
G
22
Q
U
−
G
12
Q
V
1st :
P
UV
=
P
VU
⇔
G
11
G
22
− G
12
=
−
G
11
G
22
− G
12
U
⇒
(1.182)
V
+
G
22
Q
U
−
G
11
Q
V
−
G
12
Q
U
−
G
12
Q
V
G
11
G
22
G
11
G
22
⇒
=0
.
G
12
G
12
−
−
V
U
The KL matrix equation is inverted to
Q
U
Q
V
=
G
12
−
P
U
P
V
,Q
U
=
G
12
P
U
−
1
G
11
G
22
− G
12
G
11
P
V
G
11
G
11
G
22
− G
12
,
G
22
−
G
12
Q
V
=
G
22
P
U
−
G
12
P
V
G
11
G
22
−
.
(1.183)
G
12
The inverted KL equations lead to
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