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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
3 , 1
3 , r :=
3 , 1 μν ,
M
l l (space-time)
M
{ R
}
does not allow a conformal
3 , 1 , I 4 }
,whereI 4 := [ δ μν ] := diag[1 , 1 , 1 ,
mapping to the pseudo-Euclidean manifold
{ R
1]. Note
that details referred to those authors are listed in Table 1.4 .
End of Example.
There is another interesting perspective between the
geometry of conformal mappings and the physical field
equations, say of gravitostatics, electrostatics, and magne-
tostatics. It turns out as a result of conformal field theory
that the factor of conformality , Λ 2 or λ 2 , respectively, corre-
sponds 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 Mis-
ner ( 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
A 1 ,A 1 ,A 2
to a
sphere S
r are the Universal Mercator Projection (UMP) and the Universal Stereographic Pro-
jection ( UPS ), which we are going to present to you in Example 1.11 and Fig. 1.25 ,andin
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 conformal-
ity 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
A 1 ,A 1 ,A 2 S
r .
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