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1 From Riemann manifolds to Riemann manifolds
Example 1.12 (Conformal mapping of an ellipsoid-of-revolution E
2
A 1 ,A 1 ,A 2
to a sphere S
2
r : the Universal
Stereographic Projection (UPS) of type left E
2
A 1 ,A 1 ,A 2
and right S
2
r , special Korn–Lichtenstein equations,
Cauchy–Riemann equations (d’Alembert–Euler equations)).
Let us assume that we have found a solution of the left Korn–Lichtenstein equations of the ellipsoidof-revolution E
2
A 1 ,A 1 ,A 2
parameterized by the two coordinates {Λ, Φ} which conventionally are called
{Gauss surface normal longitude, Gauss surface normal latitude}. Similarly, let us depart from a solution of the right Korn–Lichtenstein equations of the sphere S
2
r parameterized by the two coordinates
{λ, φ} which are called {spherical longitude, spherical latitude}. Here, we follow the commutative diagram of Fig. 1.26 and identify the left conformal coordinates {P, Q} with the Universal Stereographic
Projection (UPS) of E
2
A 1 ,A 1 ,A 2
, and the right conformal coordinates {p, q} with the Universal Stereographic Projection (UPS) of S
2
r , which is outlined in Box 1.31. In addition, we adopt the left and right
matrices of the metric {G l , G r } of Example 1.3.
End of Example.
We pose five problems. (i) Do the left and right conformal maps that are parameterized by
{P (Λ, Φ), Q(Λ, Φ)} and {p(λ, φ), q(λ, φ)} as “UPS left” and “UPS right” fulfil the Korn–Lichtenstein
equations, the integrability conditions (vector-valued Laplace–Beltrami equations of harmonicity, the
condition “orientation preserving conformeomorphism”? (ii) Derive the left and right factors of conformality, Λ
2 = Λ
2
1 = Λ
2
2 and λ
2 = λ
2
1 = λ
2
2 . Do the factors of conformality fulfill a special Helmholtz equation? (iii) Prove that under “UPS left” as well as “UPS right” both the ellipsoidal North Pole and the
spherical North are mapped isometrically. (iv) Derive a “simple conformal mapping” E
2
A 1 ,A 1 ,A 2
→ S
2
r .
(v) Why is the conformal mapping “UPS” called stereographic?
Fig. 1.26. Universal Polar Stereographic Projection of the sphere S
2
r , shorelines of the northern hemisphere,
Tissot ellipses of distortion. Graticule: 30
◦ in longitude, 15
◦ in latitude. Domain: {0 < λ < 2π, 0 < φ < π/2}.
1 From Riemann manifolds to Riemann manifolds
Example 1.12 (Conformal mapping of an ellipsoid-of-revolution E
2
A 1 ,A 1 ,A 2
to a sphere S
2
r : the Universal
Stereographic Projection (UPS) of type left E
2
A 1 ,A 1 ,A 2
and right S
2
r , special Korn–Lichtenstein equations,
Cauchy–Riemann equations (d’Alembert–Euler equations)).
Let us assume that we have found a solution of the left Korn–Lichtenstein equations of the ellipsoidof-revolution E
2
A 1 ,A 1 ,A 2
parameterized by the two coordinates {Λ, Φ} which conventionally are called
{Gauss surface normal longitude, Gauss surface normal latitude}. Similarly, let us depart from a solution of the right Korn–Lichtenstein equations of the sphere S
2
r parameterized by the two coordinates
{λ, φ} which are called {spherical longitude, spherical latitude}. Here, we follow the commutative diagram of Fig. 1.26 and identify the left conformal coordinates {P, Q} with the Universal Stereographic
Projection (UPS) of E
2
A 1 ,A 1 ,A 2
, and the right conformal coordinates {p, q} with the Universal Stereographic Projection (UPS) of S
2
r , which is outlined in Box 1.31. In addition, we adopt the left and right
matrices of the metric {G l , G r } of Example 1.3.
End of Example.
We pose five problems. (i) Do the left and right conformal maps that are parameterized by
{P (Λ, Φ), Q(Λ, Φ)} and {p(λ, φ), q(λ, φ)} as “UPS left” and “UPS right” fulfil the Korn–Lichtenstein
equations, the integrability conditions (vector-valued Laplace–Beltrami equations of harmonicity, the
condition “orientation preserving conformeomorphism”? (ii) Derive the left and right factors of conformality, Λ
2 = Λ
2
1 = Λ
2
2 and λ
2 = λ
2
1 = λ
2
2 . Do the factors of conformality fulfill a special Helmholtz equation? (iii) Prove that under “UPS left” as well as “UPS right” both the ellipsoidal North Pole and the
spherical North are mapped isometrically. (iv) Derive a “simple conformal mapping” E
2
A 1 ,A 1 ,A 2
→ S
2
r .
(v) Why is the conformal mapping “UPS” called stereographic?
Fig. 1.26. Universal Polar Stereographic Projection of the sphere S
2
r , shorelines of the northern hemisphere,
Tissot ellipses of distortion. Graticule: 30
◦ in longitude, 15
◦ in latitude. Domain: {0 < λ < 2π, 0 < φ < π/2}.