1-10 Two examples: Mercator Projection and Stereographic Projection 53
Note that a more elegant proof of the Korn–Lichtenstein equations based upon exterior calculus
has been presented by E. Grafarend and R. Syffus (1998d). In addition, the authors succeeded to
generalize the fundamental differential equations which govern a conformeomorphism the number of
dimensions being n (for n = 3, they coincide with the Zund equations (J. Zund (1987)) from M
3
l
to M
3
r ), namely left (pseudo-)Riemann manifold M
n
l → right (pseudo-)Riemann manifold M
n
r := E
r,s
(r + s = n). In general, conformal mappings from an arbitrary left (pseudo-)Riemann manifold M
n
l
to an arbitrary right (pseudo-)Riemann manifold M
n
r do not exist. The dimension n = 2 is just an
exception where conformal mappings always exist, though may be difficult to find. For instance, due to
involved difficulties, the Philosphical Faculty of the University of Goettingen Georgia Augusta (dated
13 June 1857) set up the “Preisaufgabe” to find a conformal mapping of the triaxial ellipsoid which had
already parameterized by C. F. Gauss in terms of “surface normal coordinates” applying the “Gauss
map”. Based upon the Jacobi’s contribution on elliptic coordinates (C. G. J. Jacobi (1839)), which
separate the Laplace–Beltrami equations of harmonicity, the “Preisschrift” of E. Schering (1857) was
finally crowned, nevertheless leaving the numerical problem open as to how to construct a conformal
map of the triaxial ellipsoid – up to now an open problem (W. Klingenberg (1982), H. Schmehl (1927),
and B. Mueller (1991)). The case of dimension n = 3 is a special case to be treated. In contrast, for
dimension n > 3, a general statement can be made: a conformeomorphism exists if and only if the
Weyl curvature tensor, being a curvature element of the Riemann curvature tensor, vanishes. We have
given in Table 1.4 a list of related, commented references. A typical example for the non-existence of
a conformeomorphism is provided by the following example.
Example 1.10 (Non-existence of a conformeomorphism).
In general relativity, the solutions of the Einstein gravitational field equations (for instance, the
Schwarzschild metric) generate a Weyl curvature different from zero. Accordingly, the space-time
pseudo-Riemann manifold M
3,1
l (space–time) → M
3,1
r
:=
R
3,1 , δ
−
µν
does not allow a conformal mapping to the pseudo-Euclidean manifold
R
3,1 , I
−
4
, where I
−
4 :=
δ
−
µν
:= diag [1, 1, 1, −1]. Note that
details referred to those authors are listed in Table 1.4.
End of Example.
Physical aside.
There is another interesting perspective between the geometry of conformal mappings and
the physical field equations, say of gravitostatics, electrostatics, and magnetostatics. It turns
out as a result of conformal field theory that the factor of conformality, Λ
2 or λ
2 , respectively, corresponds to the gravitational potential, the electric potential, and the magnetic
potential, a notion being introduced by C. F. Gauss. A highlight has been the contribution of C. W. Misner (1978) who used the vector-valued four-dimensional Laplace–Beltrami
equations (harmonic maps) as models of physical theories.
1-10 Two examples: Mercator Projection and Stereographic Projection
Two important examples for the equivalence theorem of conformal mapping, the conformal mapping from
an ellipsoid-of-revolution to the sphere: Universal Mercator Projection (UMP), Universal Stereographic
Projection (UPS).
The most famous examples for a conformal mapping of an ellipsoid-of-revolution E
2
A 1 ,A 1 ,A 2
to a sphere
S
2
r are the Universal Mercator Projection (UMP) and the Universal Stereographic Projection (UPS),
which we are going to present to you in Example 1.11 and Fig. 1.25, and in Example 1.12 and Fig. 1.26,
respectively. For both examples, we pose four problems, namely (i) prove that left and right UMP
as well as UPS fulfill the Korn–Lichtenstein equations subject to the integrability and orientation
conditions, (ii) prove that the factor of left and right conformality has to fulfill a special Helmholtz
differential equation derived from left and right Gaussian curvature, (iii) prove which coordinate line
is mapped equidistantly, and (iv) derive a “simple conformal mapping” E
2
A 1 ,A 1 ,A 2
→ S
2
r .
Note that a more elegant proof of the Korn–Lichtenstein equations based upon exterior calculus
has been presented by E. Grafarend and R. Syffus (1998d). In addition, the authors succeeded to
generalize the fundamental differential equations which govern a conformeomorphism the number of
dimensions being n (for n = 3, they coincide with the Zund equations (J. Zund (1987)) from M
3
l
to M
3
r ), namely left (pseudo-)Riemann manifold M
n
l → right (pseudo-)Riemann manifold M
n
r := E
r,s
(r + s = n). In general, conformal mappings from an arbitrary left (pseudo-)Riemann manifold M
n
l
to an arbitrary right (pseudo-)Riemann manifold M
n
r do not exist. The dimension n = 2 is just an
exception where conformal mappings always exist, though may be difficult to find. For instance, due to
involved difficulties, the Philosphical Faculty of the University of Goettingen Georgia Augusta (dated
13 June 1857) set up the “Preisaufgabe” to find a conformal mapping of the triaxial ellipsoid which had
already parameterized by C. F. Gauss in terms of “surface normal coordinates” applying the “Gauss
map”. Based upon the Jacobi’s contribution on elliptic coordinates (C. G. J. Jacobi (1839)), which
separate the Laplace–Beltrami equations of harmonicity, the “Preisschrift” of E. Schering (1857) was
finally crowned, nevertheless leaving the numerical problem open as to how to construct a conformal
map of the triaxial ellipsoid – up to now an open problem (W. Klingenberg (1982), H. Schmehl (1927),
and B. Mueller (1991)). The case of dimension n = 3 is a special case to be treated. In contrast, for
dimension n > 3, a general statement can be made: a conformeomorphism exists if and only if the
Weyl curvature tensor, being a curvature element of the Riemann curvature tensor, vanishes. We have
given in Table 1.4 a list of related, commented references. A typical example for the non-existence of
a conformeomorphism is provided by the following example.
Example 1.10 (Non-existence of a conformeomorphism).
In general relativity, the solutions of the Einstein gravitational field equations (for instance, the
Schwarzschild metric) generate a Weyl curvature different from zero. Accordingly, the space-time
pseudo-Riemann manifold M
3,1
l (space–time) → M
3,1
r
:=
R
3,1 , δ
−
µν
does not allow a conformal mapping to the pseudo-Euclidean manifold
R
3,1 , I
−
4
, where I
−
4 :=
δ
−
µν
:= diag [1, 1, 1, −1]. Note that
details referred to those authors are listed in Table 1.4.
End of Example.
Physical aside.
There is another interesting perspective between the geometry of conformal mappings and
the physical field equations, say of gravitostatics, electrostatics, and magnetostatics. It turns
out as a result of conformal field theory that the factor of conformality, Λ
2 or λ
2 , respectively, corresponds to the gravitational potential, the electric potential, and the magnetic
potential, a notion being introduced by C. F. Gauss. A highlight has been the contribution of C. W. Misner (1978) who used the vector-valued four-dimensional Laplace–Beltrami
equations (harmonic maps) as models of physical theories.
1-10 Two examples: Mercator Projection and Stereographic Projection
Two important examples for the equivalence theorem of conformal mapping, the conformal mapping from
an ellipsoid-of-revolution to the sphere: Universal Mercator Projection (UMP), Universal Stereographic
Projection (UPS).
The most famous examples for a conformal mapping of an ellipsoid-of-revolution E
2
A 1 ,A 1 ,A 2
to a sphere
S
2
r are the Universal Mercator Projection (UMP) and the Universal Stereographic Projection (UPS),
which we are going to present to you in Example 1.11 and Fig. 1.25, and in Example 1.12 and Fig. 1.26,
respectively. For both examples, we pose four problems, namely (i) prove that left and right UMP
as well as UPS fulfill the Korn–Lichtenstein equations subject to the integrability and orientation
conditions, (ii) prove that the factor of left and right conformality has to fulfill a special Helmholtz
differential equation derived from left and right Gaussian curvature, (iii) prove which coordinate line
is mapped equidistantly, and (iv) derive a “simple conformal mapping” E
2
A 1 ,A 1 ,A 2
→ S
2
r .