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C μν = λ 2 δ μν or C r = λ 2 I 2 .
(D.4)
Consequently, the left symmetric metric form ( D.5 ) enjoys a particular structure which is called
conformally flat . The factor of conformality λ 2 is generated by the simultaneous diagonal-
ization of the matrix pair
{
C r , G r }
, namely from the general eigenvalue-eigenvector problem
λ 2 G r ) F r = 0, the characteristic equation
λ 2 G r
= 0 subject to G r =I 2 and the
canonical conformality postulate λ 1 = λ 2 = λ 2 . The condition of conformality transforms the
right Cauchy-Green deformation tensor into ( D.6 ).
(C r
|
C r
|
d S 2 = λ 2 (d x 2 +d y 2 ) ,
(D.5)
1
λ 2 I 2 .
C r =J r G l J r = λ 2 I 2 or C r =J l G J J l =
(D.6)
Equation ( D.6 ) can be interpreted as an orthogonality condition of the rows of the left Jacobi
matrix with respect to the inverse left metric matrix G l .G l -orthogonality of the rows of the
left Jacobi matrix J l implies ( D.7 ). Equation ( D.7 ) be derived from ( D.8 ), namely with respect
to the permutation symbol ( D.9 ).
d x =
d y,
(D.7)
d x = x U d U + x V d V = x I d U I ,
x 1 := ∂x
∂U , 2 := ∂x
∂V ,
d y := e IJ det[G l ]G JK y K d U I
∀{I,J,K}∈{ 1 , 2 },
(D.8)
y 1 := ∂y
∂U , 2 := ∂y
∂V ,
+1 even permutation of the indices
e IJ =
1 odd permutation of the indices ,
0 th rw e
(D.9)
d x := x I d U I =
= e IJ det[G l ]G JK y K d U I =
∂U I = e IJ det[G l ]G JK ∂y
∂x
d y
∂U K .
(D.10)
Indeed, we have to take advantage of the Hodge star operator , which generalizes the cross product
on R
3 . Lemma D.1 outlines the definition of the Hodge star operator of a one-differential form. It
may be a surprise that ( D.10 ) constitutes the Korn-Lichtenstein equations of a conformal mapping
parameterized by
. Note the representation ( D.11 ) of the inverse left metric
matrix in order to derive the third version of the Korn-Lichtenstein equations in Lemma D.1 .
G l = G IJ = G 11
{
x ( U, V ) ,y ( U, V )
}
G 12
G 21
G 22
(subject to G 21 = G 12 )
(D.11)
 
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