From Riemann Manifolds to Euclidean Manifolds - Map Projections: Cartographic Information Systems - page 118

Box 2.2 summarizes the operational procedure for generating a conformal diffeomorphism, also
called conformeomorphism ,

3 , again in terms of exterior calculus. First,
we introduce the differential one-forms, the differential two-forms, and the differential three-
forms. Second, we apply the Hodge star operator (i) to

l r = 2 ,δ μν }

M → M { R = E

∗ d x etc., (ii) to ∗ (d y ∧ d z ) etc., and

d z ). The columns [ x 1 ,x 2 ,x 3 ] T , [ y 1 ,y 2 ,y 3 ] T ,and[ z 1 ,z 2 ,z 3 ] T may be
considered orthogonal. Third, we represent the expression

(iii) to ∗ (d y ∧ d y ∧

∗ (d y ∧ d z ) as an example explicitly.

3 × 3 × 3 ( L, M 1 ,M 2 ∈{

Again, the three-dimensional permutation symbol e LM 1 M 2 ∈ R 1 , 2 } )asa

three-dimensional array is defined. Fourth, we explicitly compute the expression d x = ∗ (d y ∧ d) z ,

3
l

3 3 : compare with

the Zund equations of a three-dimensional conformal mapping M → M

r =

E

Lemma 2.5 .

3
l

r = 3 ,δ μν 3 , exterior calculus).

Box 2.2 (Conformal diffeomorphism M → M { R } = E

Differential frame:
d x = x 1 d U + x 2 d V + x 3 d W
(i) d y = y 1 d U + y 2 d V + y 3 d W
d z = z 1 dU + z 2 dV + z 3 d W

⎤

⎦ (one-forms) ,

(2.34)

(ii) d y ∧ d z, d z ∧ d x, d x ∧ d y, (two-forms) ,

(iii) d x ∧ d y ∧ d z (three-form) .

Hodge star operator:
(i) ∗ d x =d y ∧ d z,

∗ d y =d z ∧ d x, ∗ d=d x ∧ d y ;

(ii) ∗ (d y ∧ d z )=d x, ∗ (d z ∧ d x )=d y, ∗ (d x ∧ d y )=d z ; (2.35)

(iii) ∗ (d x ∧ d y ∧ d z )=1 .

Example :

∀ L, M 1 ,M 2 ,N 1 ,N 2 ∈{ 1 , 2 , 3 } :

3

e LM 1 M 2

∂y
∂U N 1

∂z

G M 1 N 1 G M 2 N 2

∂U N 2 d U L .

∗ (d y ∧ d z )= | G l | (2.36)

L,M 1 ,M 2 ,N 1 ,N 2 =1

Permutation symbol:

⎧
⎨

+1 for an even permutation of the indices L, M 1 ,M 2 ∈{ 1 , 2 , 3 }

e LM 1 M 2 = − 1 for an odd permutation of the indices L, M 1 ,M 2 ∈{ 1 , 2 , 3 } . (2.37)

⎩

0 th rw e

Zund equations of a two-dimensional conformal diffeomorphism
in exterior calculus:

3

x M d U M =

d x =

M =1

3

e LM 1 M 2

∂y
∂U N 1

∂z

G M 1 N 1 G M 2 N 2

∂U N 2 d U L =

= | G l | ∗ (d y ∧ d z ) (2.38)

L,M 1 ,M 2 ,N 1 ,N 2 =1