Korn–Lichtenstein and d’Alembert–Euler Equations - Map Projections: Cartographic Information Systems - page 701

D
Korn-Lichtenstein and d'Alembert-Euler Equations

Conformal mapping, Korn-Lichtenstein equations and d'Alembert-Euler (Cauchy-Riemann)
equations. Polynomial solutions. Conformeomorphism, condition of conformality.

D-1 Korn-Lichtenstein Equations

Our starting point for the construction of a conformal diffeomorphism (in short conformeomor-
phism )isprovidedby( D.1 ) of a left two-dimensional Riemann manifold { M

2 , G l } subject to the

left metric G l = { G MN } parameterized by the left coordinates { U, V } = { U 1 ,U 2 } and a right

two-dimensional Euclidean manifold { M

2 , G r =I 2 }

subject to the right metric G r =I 2 =diag[1,

x 1 ,x 2

1] or { g μν } = { δ μν } parameterized by the right coordinates { x, y } = { } of Cartesian type.

d x
d y

=J l d U , J l = x U x V .

(D.1)

d V

y U y V

J l constitutes the left Jacobi matrix of first partial derivatives, which is related to the right
Jacobi matrix J r by means of the duality J l J r =I 2 or J l =J − r and J r =J − l with J l ∈ R

2 × 2 and

J r ∈ R

2 × 2 . Let us compare the left and the right symmetric metric forms, the squared infinitesimal
arc lengths ( D.2 ) subject to the summation convention over repeated indices which run here
from one to two, i.e.

.Equation( D.3 ) constitutes the right Cauchy-Green
deformation tensor which has to be constraint to the condition of conformality ( D.4 ).

{ M,N,μ,ν }∈{ 1 . 2 }

d S 2 =d U M G MN d U N =d x μ ∂U M

∂x μ G MN ∂U M

=d s 2 =d x 2 +d y 2 =

∂x ν

versus (D.2)

=d x μ C μν d x ν =d x μ δ μν d x ν ,

C μν := ∂U M

∂x μ G MN ∂U N

or C r := J r G l J r ,

(D.3)

∂x ν