102 2 From Riemann manifolds to Euclidean manifolds
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 M
2
l → M
2
r := {R
2 , δ µν } = E
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 M
3
l → M
3
r := {R
3 , δ µν } = E
3 by means of exterior calculus
taking advantage of the Hodge star operator in R
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 R
3 . Box 2.1 summarizes the various steps to produce a conformal
diffeomorphism M
2
l → M
2
r = {R
2 , δ µν } = E
2 in terms of exterior calculus. First, we introduce the left
Jacobi map {dx, dy} → {dU, dV } and the right Jacobi map {dU, dV } → {dx, dy}. Second, we compute
the right Cauchy–Green matrix C r subject to its conformal structure C r = λ
2 I 2 and C
−1
r = λ
−2 I 2 . We
are led to a representation of the conformal right Cauchy–Green matrix C r = J
T
r G l J r = λ
2 I 2 or
C
−1
r = J
T
l G
−1
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
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-byone, we define dx, x 1 , x 2 , and dy
∗ . 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 dx = dy
∗ of the
Korn–Lichtenstein equations: compare with Lemma 2.4.
Lemma 2.4 (E. Grafarend and R. Syffus (1998d, p. 292), conformeomorphism M
2
l → M
2
r := {R
2 , δ µν },
Korn–Lichtenstein equations).
The following formulations of the Korn–Lichtenstein equations producing a conformal diffeomorphism
M
2
l → M
2
r := {R
2 , δ µν } are equivalent.
Formulation (i):
dx = ∗dy .
(2.23)
Formulation (ii):
∂x
∂U L = e LM
det [G l ]G
MN ∂y
∂U N .
(2.24)
Formulation (iii):
x U =
1
|G l |
(−G 12 y U + G 11 y V ) , x V =
1
|G l |
(−G 22 y U + G 12 y V ) ,
(2.25)
G l = [G MN ] =
⎡
⎣
G 11 G 12
G 12 G 22
⎤
⎦ ⇔
1
|G l |
⎡
⎣
G 22 −G 12
−G 12 G 11
⎤
⎦ =
G
LM
= G
−1
l
,
(2.26)
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.
2-32 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 M
2
l → M
2
r := {R
2 , δ µν } = E
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 M
3
l → M
3
r := {R
3 , δ µν } = E
3 by means of exterior calculus
taking advantage of the Hodge star operator in R
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 R
3 . Box 2.1 summarizes the various steps to produce a conformal
diffeomorphism M
2
l → M
2
r = {R
2 , δ µν } = E
2 in terms of exterior calculus. First, we introduce the left
Jacobi map {dx, dy} → {dU, dV } and the right Jacobi map {dU, dV } → {dx, dy}. Second, we compute
the right Cauchy–Green matrix C r subject to its conformal structure C r = λ
2 I 2 and C
−1
r = λ
−2 I 2 . We
are led to a representation of the conformal right Cauchy–Green matrix C r = J
T
r G l J r = λ
2 I 2 or
C
−1
r = J
T
l G
−1
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
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-byone, we define dx, x 1 , x 2 , and dy
∗ . 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 dx = dy
∗ of the
Korn–Lichtenstein equations: compare with Lemma 2.4.
Lemma 2.4 (E. Grafarend and R. Syffus (1998d, p. 292), conformeomorphism M
2
l → M
2
r := {R
2 , δ µν },
Korn–Lichtenstein equations).
The following formulations of the Korn–Lichtenstein equations producing a conformal diffeomorphism
M
2
l → M
2
r := {R
2 , δ µν } are equivalent.
Formulation (i):
dx = ∗dy .
(2.23)
Formulation (ii):
∂x
∂U L = e LM
det [G l ]G
MN ∂y
∂U N .
(2.24)
Formulation (iii):
x U =
1
|G l |
(−G 12 y U + G 11 y V ) , x V =
1
|G l |
(−G 22 y U + G 12 y V ) ,
(2.25)
G l = [G MN ] =
⎡
⎣
G 11 G 12
G 12 G 22
⎤
⎦ ⇔
1
|G l |
⎡
⎣
G 22 −G 12
−G 12 G 11
⎤
⎦ =
G
LM
= G
−1
l
,
(2.26)
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.