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C
μν
=
λ
2
δ
μν
or C
r
=
λ
2
I
2
.
(D.4)
Consequently, the left symmetric metric form (
D.5
) enjoys a particular structure which is called
conformally flat
. The factor of conformality
λ
2
is generated by the simultaneous diagonal-
ization of the matrix pair
{
C
r
,
G
r
}
, namely from the general eigenvalue-eigenvector problem
λ
2
G
r
)
F
r
= 0, the characteristic equation
λ
2
G
r
= 0 subject to G
r
=I
2
and the
canonical conformality postulate
λ
1
=
λ
2
=
λ
2
. The condition of conformality transforms the
right Cauchy-Green deformation tensor into (
D.6
).
(C
r
−
|
C
r
−
|
d
S
2
=
λ
2
(d
x
2
+d
y
2
)
,
(D.5)
1
λ
2
I
2
.
C
r
=J
r
G
l
J
r
=
λ
2
I
2
or C
−
r
=J
l
G
−
J
J
l
=
(D.6)
Equation (
D.6
) can be interpreted as an orthogonality condition of the rows of the left Jacobi
matrix with respect to the inverse left metric matrix G
−
l
.G
−
l
-orthogonality of the rows of the
left Jacobi matrix J
l
implies (
D.7
). Equation (
D.7
) be derived from (
D.8
), namely with respect
to the permutation symbol (
D.9
).
d
x
=
∗
d
y,
(D.7)
d
x
=
x
U
d
U
+
x
V
d
V
=
x
I
d
U
I
,
x
1
:=
∂x
∂U
,
2
:=
∂x
∂V
,
∗
d
y
:= e
IJ
det[G
l
]G
JK
y
K
d
U
I
∀{I,J,K}∈{
1
,
2
},
(D.8)
y
1
:=
∂y
∂U
,
2
:=
∂y
∂V
,
⎧
⎨
+1 even permutation of the indices
−
e
IJ
=
1 odd permutation of the indices
,
0 th rw e
(D.9)
⎩
d
x
:=
x
I
d
U
I
=
=
e
IJ
det[G
l
]G
JK
y
K
d
U
I
=
∂U
I
=
e
IJ
det[G
l
]G
JK
∂y
∂x
∗
d
y
⇔
∂U
K
.
(D.10)
Indeed, we have to take advantage of the
Hodge star operator
, which generalizes the cross product
on
R
3
. Lemma
D.1
outlines the definition of the Hodge star operator of a one-differential form. It
may be a surprise that (
D.10
) constitutes the Korn-Lichtenstein equations of a conformal mapping
parameterized by
. Note the representation (
D.11
) of the inverse left metric
matrix in order to derive the third version of the Korn-Lichtenstein equations in Lemma
D.1
.
G
l
=
G
IJ
=
G
11
{
x
(
U, V
)
,y
(
U, V
)
}
G
12
G
21
G
22
(subject to
G
21
=
G
12
)
⇔
(D.11)
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