1-9 Equivalence theorem of conformal mapping 43
1-9 Equivalence theorem of conformal mapping
“Experience proves that anyone who studied geometry is infinitely
quicker to grasp difficult subjects than one who has not.”
(Plato. The Republic Book 7, 375 B. C.)
The equivalence theorem of conformal mapping from the left to the right two-dimensional Riemann
manifold (conformeomorphism), generalized Korn–Lichtenstein equations.
We shall define conformeophism as well as angular shear, and shall present the equivalence theorem
that relates conformeomorphism to a special structure of the Cauchy–Green deformation tensor, the
Euler–Lagrange deformation tensor, the left and right principal stretches (left and right eigenvalues)
as well as dilatations, before we are led to the generalized Korn–Lichtenstein equations which govern
any conformal mapping: compare with Definition 1.10 and Theorem 1.11. For a further motivation,
we refer to Fig. 1.22, which presents an image of Lichtenstein’s original publication “Zur Theorie der
konformen Abbildung”.
Definition 1.10 (Conformal mapping).
An orientation preserving diffeomorphism f : M
2
l → M
2
r is called angle preserving conformal mapping
(conformeomorphism, inner product preserving) if Ψ l = Ψ r and Σ l = Σ = 0 ⇔ Σ r = σ = 0 for all
points of M
2
l and M
2
r , respectively, holds.
End of Definition.
Theorem 1.11 (Conformeomorphism M
2
l → M
2
r , conformal mapping).
Let f : M
2
l → M
2
r be an orientation preserving conformal mapping. Then the following conditions
(i)–(iv) are equivalent:
(i) Ψ l ( ˙
U 1 , ˙
U 2 ) = Ψ r ( ˙
u 1 , ˙
u 2 ) ,
(1.153)
for all tangent vectors
˙
U 1 , ˙
U 2
and their images
˙
u 1 , ˙
u 2
, respectively;
(ii) C l = Λ
2 (U 0 )G l , C l G
−1
l
= Λ
2 (U 0 )I 2 versus C r = λ
2 (u 0 )G r , C r G
−1
r = λ
2 (u 0 )I 2 ,
E l = K(U 0 )G l , E l G
−1
l
= K(U 0 )I 2 versus E r = κ(u 0 )G r , E r G
−1
r = κ(u 0 )I 2 ;
(1.154)
(iii) K = (Λ
2
− 1)/2 , Λ
2 = 2K + 1 versus (λ
2
− 1)/2 = κ , 2κ + 1 = λ
2 ,
Λ 1 = Λ 2 = Λ(U 0 )
v e r s u s
λ 1 = λ 2 = λ(u 0 ) ,
K 1 = K 2 = K(U 0 )
v e r s u s
κ 1 = κ 2 = κ(u 0 ) ,
Λ
2 (U 0 ) =
1
2
tr
C l G
−1
l
versus
λ
2 (u 0 ) =
1
2
tr
C r G
−1
r
;
(1.155)
left dilatation:
right dilatation:
K =
1
2
tr
E l G
−1
l
versus
κ =
1
2
tr
E r G
−1
r
,
tr
C l G
−1
l
= 2
det
C l G
−1
l
versus tr
C r G
−1
r
= 2
det
C r G
−1
r
,
tr
E l G
−1
l
= 2
det
E l G
−1
l
versus tr
E r G
−1
r
= 2
det
E r G
−1
r
;
(1.156)
(iv) generalized Korn–Lichtenstein equations (special case: g 12 = 0):
u U
u V
=
1
G 11 G 22 − G 2
12
g 11
g 22
−G 12 G 11
−G 22 G 12
v U
v V
,
(1.157)
subject to the integrability conditions u UV = u V U and v UV = v V U .
End of Theorem.