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subject to the integrability conditions
∂
2
x
∂U∂V
=
∂
2
x
∂V ∂U
,
∂
2
y
∂U∂V
=
∂
2
y
∂V ∂U
.
(2.27)
End of Lemma.
2
l
r
=
2
,δ
μν
2
, exterior calculus).
Box 2.1 (Conformal diffeomorphism
M
→
M
{
R
}
=
E
Diffeomorphism :
d
x
d
y
=J
l
d
U
d
U
d
V
=J
r
d
x
or
d
V
d
y
⇔
J
l
=J
−
1
(2.28)
r
⇔
J
r
=J
−
l
.
Right Cauchy-Green matrix for a conformal diffeomorphism:
C
r
=J
r
G
l
J
r
=
λ
2
I
2
(2.29)
⇔
C
−
r
=J
l
G
−
l
J
l
=
λ
−
2
I
2
.
The rows of the left Jacobi matrix are G
−
l
orthogonal :
2
x
M
d
U
M
,
1
:=
D
U
x
=
x
U
,
2
:=
D
V
x
=
x
V
.
d
x
=
x
U
d
U
+
x
d
V
d
V
=
(2.30)
M
=1
Hodge star operator:
2
e
LM
det [G
l
]
G
MN
y
N
d
U
L
,
∗
d
y
:=
(2.31)
L,M,N
=1
subject to
y
1
:=
D
Uy
=
y
U
,
2
:=
D
Vy
=
y
V
.
Permutation symbol:
⎧
⎨
+1 for an even permutation of the indices
L, M
∈{
1
,
2
}
e
LM
=
−
1 for an odd permutation of the indices
L, M
∈{
1
,
2
}
.
(2.32)
⎩
0 th rw e
Korn-Lichtenstein equations in exterior calculus:
2
2
e
LM
det [G
l
]
G
MN
y
N
d
U
L
=d
y
∗
x
M
d
U
M
=
d
x
=
M
=1
L,M,N
=1
⇔
(2.33)
∂U
L
=
e
LM
det [G
l
]
G
MN
∂y
∂x
∂U
N
,
d
x
=d
y
∗
.
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