From Riemann Manifolds to Riemann Manifolds - Map Projections: Cartographic Information Systems - page 54

g 22

1
G 11 G 22 − G 12 ⇒

g 11 x 2 .
The converse (iv) ⇒ (ii) is obvious.

x 1 =G l A

End of Proof (third part).

Here is the remark relating to G l being diagonal, not unity, of course. An obvious generalization
for solving ( γ )and( δ )for g 12 = 0 would be the

“Ansatz x 1 =G l X x 2 , X= yx

,

(1.168)

− x − y

the superposition of a diagonal trace-free matrix diag [ y, y ] and an antisymmetric matrix Ax.

Indeed, we succeed in determining the unknowns x and y according to the above steps, but fail
to arrive at linear relations between the partials

−

{ u U ,u V ,v U ,v V } .

M l → M l

Fig. 1.24. Flow chart, conformal mapping

r has been chosen.
In Fig. 1.24 , the alternative path of generating a conformal mapping from a left curved surface
to a right curved surface is outlined. First, the original coordinates

In practice, a different way in constructing a conformeomorphism M

l

→ M

{ U 1 ,U 2 } or { U, V } ,which

parameterize the left surface, are transformed to alternative left conformal coordinates { P,Q } .

d S 2 =
Λ 2 (d P 2 +d Q 2 ) is described by identical metric coecients G PP = G QQ = Λ 2 and G PQ =0.

which are also called isometric or isothermal . Indeed, the left differential invariant I l ∼