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(iv) (Generalized Korn-Lichtenstein equations, Cauchy-Riemann equations, subject to the
integrability conditions u UV = u VU and v UV = v VU )
u U
u V
=
u U
u V
.
1
G 11 G 22
G 12 G 11
(2.22)
G 22 G 12
G 12
End of Lemma.
2-32 Higher-Dimensional Conformal Mapping
In order to develop the theory of a higher-dimensional conformal diffeomorphism (in Gauss's
words: “in kleinsten Teilen ahnlich”), we first derive the Korn-Lichtenstein equations of a two-
dimensional conformal mapping
2 by means of exterior calculus ,
namely by means of the Hodge star operator . With such an experience built up, second, we derive
the Zund equations of a three-dimensional conformal mapping
2
l
2
2 μν }
M
M
r :=
{ R
=
E
3
l
3
3 μν }
3
M
M
r :=
{ R
=
E
3 . Note that the
Hodge star operator generalizes the vector product , also called cross product or outer product ,to
any dimension. Indeed, the classical vector product serves us only in
by means of exterior calculus taking advantage of the Hodge star operator in
R
3 .Box 2.1 summarizes
R
2
l
r =
2 μν
2 in terms
the various steps to produce a conformal diffeomorphism
M
M
{ R
}
=
E
of exterior calculus. First, we introduce the left Jacobi map
and the
right Jacobi map { d U, d V }→{ d x, d y} . Second, we compute the right Cauchy-Green matrix C r
subject to its conformal structure C r = λ 2 I 2 and C r = λ 2 I 2 . We are led to a representation of the
conformal right Cauchy-Green matrix C r =J r G l J r = λ 2 I 2 or C r =J l G l J l = λ 2 I 2 in terms
of the Jacobi matrices J l and J r . The rows of the left Jacobi matrix can be interpreted as “G 1
{
d x, d y
}→{
d U, d V
}
l
orthogonal”, while the right Jacobi matrix can be interpreted as “G l orthogonal”. Third, this result
of conformal geometry is used by the Hodge star operator. One-by-one, we define d x, x 1 ,x 2 ,and
d y . Here, we make use of the two-dimensional permutation symbol e LM R
2 × 2 ( L, M ∈{ 1 , 2 } ).
Fourth, we explicitly represent the exterior form d x =d y of the Korn-Lichtenstein equations:
compare with Lemma 2.4 .
2
l
2
2 μν
Lemma 2.4 ( Grafarend and Syffus ( 1998d , p. 292), conformeomorphism
M
M
r :=
{ R
}
,
Korn-Lichtenstein equations).
The following formulations of the Korn-Lichtenstein equations producing a conformal diffeomor-
phism
2
l
2
2 μν }
M
M
r :=
{ R
are equivalent.
Formulation (i):
d x =
d y.
(2.23)
Formulation (ii):
∂U L = e LM det [G l ] G MN
∂x
∂y
∂U N .
(2.24)
Formulation (iii):
1
|
1
|
x U =
(
G 12 yU + G 11 yV ) , V =
(
G 22 yU + G 12 yV ) ,
(2.25)
G l |
G l |
G l =[ G MN ]= G 11 G 12
G 22
=[ G LM ]=G l ,
1
G 12
(2.26)
G 12 G 22
G 12 G 11
|
G l |
 
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