1-9 Equivalence theorem of conformal mapping 51
The proof of the operational theorem of conformal mapping M
2
l → M
2
r rests upon the existence
theorem of S. S. Cherne (1955), where it is shown that under rather mild certainty assumptions, namely
C
2,x , conformal coordinates (isometric coordinates, isothermal coordinates) exist as solutions of the
left or right Korn–Lichtenstein equations. Let us here also refer to the following authors. W. Blaschke
and K. Leichtweiß (1973), D. G. L. Boulware, L. S. Brown and R. D. Peccei (1970), J. P. Bourguignon
(1970), B. Y. Chen (1973), B. Y. Chen and K. Yano (1973), S. S. Chern (1967), S. S. Chern, P. Hartman
and A. Wintner (1954), M. Do Carmo, M. Dajczer and F. Mercuri (1985), L. P. Eisenhart (1949),
L. Euler (1755, 1777a), S. Ferrara, A. F. Grillo and R. Gatto (1972), A. Finzi (1922), C. F. Gauss
(1822, 1844), H. Goenner, E. Grafarend and R. J. You (1994), C. G. J. Jacobi (1839), S. Heitz (1988),
W. Klingenberg (1982), K. K¨ onig and K. H. Weise (1951), A. Korn (1914), L. Krueger (1903, 1922),
N. Kuiper (1949, 1950), R. S. Kulkarni (1969, 1972), R. S. Kulkarni and U. Pinkall (1988), J. Lafontaine
(1988), J. L. Lagrange (1781), G. M. Lancaster (1969, 1973), L. Lichtenstein (1911, 1916), J. Liouville (1850), A. I. Markuschewitsch (1955), L. Mirsky (1960), C. W. Misner (1978), S. K. Mitra and
C. R. Rao (1968), B. Moor and H. Zha (1991), J. D. Moore (1977), S. Nishikawa (1974), G. Ricci (1918),
B. Riemann (1851), H. Samelson (1969), E. Schering (1857), H. Schmehl (1927), R. Schoen (1984),
J. A. Schouten (1921), M. Spivak (1979), E. M. Stein and G. Weiss (1968), H. Weber (1867), H. Weyl
(1918, 1921), T. Wray (1974), K. Yano (1970), A. I. Yanushauskas (1982), M. Zadro and A. Carminelli
(1966), and J. Zund (1987).
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 coordinates
{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 . Up to the 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 V U and
Q UV = Q V U are converted to the Laplace–Beltrami equation.
KL, 1st equation and 2nd equation, lead to
P UV =
−G 12 Q U +G 11 Q V
√
G 11 G 22 −G 2
12
V
, P V U =
−G 22 Q U +G 12 Q V
√
G 11 G 22 −G 2
12
U
.
(1.181)
1st: P UV = P V U ⇔
G 11 Q V −G 12 Q U
√
G 11 G 22 −G 2
12
V
= −
G 22 Q U −G 12 Q V
√
G 11 G 22 −G 2
12
U
⇒
⇒
G 11 Q V −G 12 Q U
√
G 11 G 22 −G 2
12
V
+
G 22 Q U −G 12 Q V
√
G 11 G 22 −G 2
12
U
= 0 .
(1.182)
The KL matrix equation is inverted to
Q U
Q V
=
1
√
G 11 G 22 −G 2
12
G 12 −G 11
G 22 −G 12
P U
P V
, Q U =
G 12 P U −G 11 P V
√
G 11 G 22 −G 2
12
, Q V =
G 22 P U −G 12 P V
√
G 11 G 22 −G 2
12
. (1.183)
The inverted KL equations lead to
Q UV = −
G 11 P V −G 12 P U
√
G 11 G 22 −G 2
12
V
, Q V U =
G 22 P U −G 12 P V
√
G 11 G 22 −G 2
12
U
.
(1.184)
2nd: Q UV = Q V U ⇔ −
G 11 P V −G 12 P U
√
G 11 G 22 −G 2
12
V
=
G 22 P U −G 12 P V
√
G 11 G 22 −G 2
12
U
⇒
⇒
G 11 P V −G 12 P U
√
G 11 G 22 −G 2
12
V
+
G 22 P U −G 12 P V
√
G 11 G 22 −G 2
12
U
= 0 .
(1.185)
End of Proof (the first step).
The proof of the operational theorem of conformal mapping M
2
l → M
2
r rests upon the existence
theorem of S. S. Cherne (1955), where it is shown that under rather mild certainty assumptions, namely
C
2,x , conformal coordinates (isometric coordinates, isothermal coordinates) exist as solutions of the
left or right Korn–Lichtenstein equations. Let us here also refer to the following authors. W. Blaschke
and K. Leichtweiß (1973), D. G. L. Boulware, L. S. Brown and R. D. Peccei (1970), J. P. Bourguignon
(1970), B. Y. Chen (1973), B. Y. Chen and K. Yano (1973), S. S. Chern (1967), S. S. Chern, P. Hartman
and A. Wintner (1954), M. Do Carmo, M. Dajczer and F. Mercuri (1985), L. P. Eisenhart (1949),
L. Euler (1755, 1777a), S. Ferrara, A. F. Grillo and R. Gatto (1972), A. Finzi (1922), C. F. Gauss
(1822, 1844), H. Goenner, E. Grafarend and R. J. You (1994), C. G. J. Jacobi (1839), S. Heitz (1988),
W. Klingenberg (1982), K. K¨ onig and K. H. Weise (1951), A. Korn (1914), L. Krueger (1903, 1922),
N. Kuiper (1949, 1950), R. S. Kulkarni (1969, 1972), R. S. Kulkarni and U. Pinkall (1988), J. Lafontaine
(1988), J. L. Lagrange (1781), G. M. Lancaster (1969, 1973), L. Lichtenstein (1911, 1916), J. Liouville (1850), A. I. Markuschewitsch (1955), L. Mirsky (1960), C. W. Misner (1978), S. K. Mitra and
C. R. Rao (1968), B. Moor and H. Zha (1991), J. D. Moore (1977), S. Nishikawa (1974), G. Ricci (1918),
B. Riemann (1851), H. Samelson (1969), E. Schering (1857), H. Schmehl (1927), R. Schoen (1984),
J. A. Schouten (1921), M. Spivak (1979), E. M. Stein and G. Weiss (1968), H. Weber (1867), H. Weyl
(1918, 1921), T. Wray (1974), K. Yano (1970), A. I. Yanushauskas (1982), M. Zadro and A. Carminelli
(1966), and J. Zund (1987).
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 coordinates
{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 . Up to the 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 V U and
Q UV = Q V U are converted to the Laplace–Beltrami equation.
KL, 1st equation and 2nd equation, lead to
P UV =
−G 12 Q U +G 11 Q V
√
G 11 G 22 −G 2
12
V
, P V U =
−G 22 Q U +G 12 Q V
√
G 11 G 22 −G 2
12
U
.
(1.181)
1st: P UV = P V U ⇔
G 11 Q V −G 12 Q U
√
G 11 G 22 −G 2
12
V
= −
G 22 Q U −G 12 Q V
√
G 11 G 22 −G 2
12
U
⇒
⇒
G 11 Q V −G 12 Q U
√
G 11 G 22 −G 2
12
V
+
G 22 Q U −G 12 Q V
√
G 11 G 22 −G 2
12
U
= 0 .
(1.182)
The KL matrix equation is inverted to
Q U
Q V
=
1
√
G 11 G 22 −G 2
12
G 12 −G 11
G 22 −G 12
P U
P V
, Q U =
G 12 P U −G 11 P V
√
G 11 G 22 −G 2
12
, Q V =
G 22 P U −G 12 P V
√
G 11 G 22 −G 2
12
. (1.183)
The inverted KL equations lead to
Q UV = −
G 11 P V −G 12 P U
√
G 11 G 22 −G 2
12
V
, Q V U =
G 22 P U −G 12 P V
√
G 11 G 22 −G 2
12
U
.
(1.184)
2nd: Q UV = Q V U ⇔ −
G 11 P V −G 12 P U
√
G 11 G 22 −G 2
12
V
=
G 22 P U −G 12 P V
√
G 11 G 22 −G 2
12
U
⇒
⇒
G 11 P V −G 12 P U
√
G 11 G 22 −G 2
12
V
+
G 22 P U −G 12 P V
√
G 11 G 22 −G 2
12
U
= 0 .
(1.185)
End of Proof (the first step).