104 2 From Riemann manifolds to Euclidean manifolds
Box 2.2 (Conformal diffeomorphism M
3
l → M
3
r = {R
3 , δ µν } = E
3 , exterior calculus).
Differential frame:
(i)
dx = x 1 dU + x 2 dV + x 3 dW
dy = y 1 dU + y 2 dV + y 3 dW
dz = z 1 dU + z 2 dV + z 3 dW
3
7
7
7
5
(one-forms) ,
(ii) dy ∧ dz , dz ∧ dx , dx ∧ dy , (two-forms) ,
(iii) dx ∧ dy ∧ dz
(three-form) .
(2.34)
Hodge star operator:
(i) ∗dx = dy ∧ dz , ∗dy = dz ∧ dx , ∗dz = dx ∧ dy ;
(ii) ∗(dy ∧ dz) = dx , ∗(dz ∧ dx) = dy , ∗(dx ∧ dy) = dz ;
(iii) ∗ (dx ∧ dy ∧ dz) = 1 .
(2.35)
Example:
∀ L, M 1 , M 2 , N 1 , N 2 ∈ {1, 2, 3} :
∗(dy ∧ dz) =
3
X
L,M 1 ,M 2 ,N 1 ,N 2 =1
e LM 1 M 2
p |G l |G
M 1 N 1 G
M 2 N 2 ∂y
∂U N 1
∂z
∂U N 2
dU
L .
(2.36)
Permutation symbol:
e LM 1 M 2 =
8
<
:
+1 for an even permutation of the indices L, M 1 , M 2 ∈ {1, 2, 3}
−1 for an odd permutation of the indices L, M 1 , M 2 ∈ {1, 2, 3}
0
otherwise
.
(2.37)
Zund equations of a two-dimensional conformal diffeomorphism in exterior calculus:
dx =
3
X
M=1
x M dU
M =
=
3
X
L,M 1 ,M 2 ,N 1 ,N 2 =1
e LM 1 M 2
p |G l |G
M 1 N 1 G
M 2 N 2 ∂y
∂U N 1
∂z
∂U N 2
dU
L = ∗(dy ∧ dz)
⇔
∂x
∂U L = e LM 1 M 2
p
det [G l ]G
M 1 N 1 G
M 2 N 2 ∂y
∂U N 1
∂z
∂U N 2
,
dx = ∗(dy ∧ dz) .
(2.38)