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2sin
φ
+sin
2
φ
cos
φ
1
2
1
cos
φ
cos
2
φ
+1
−
1
cos
φ
1
−
sin
φ
cos
φ
=
−
=
−
=
cos
φ
tan
π
.
1
Φ
2
−
4
−
Solution (the first problem).
Start from the conformal map defined in Box
1.31
. The three conditions to be fulfilled are given
in Box
1.32
for “left UPS” and in Box
1.33
for
“right
UP
S”. First
, we sp
ecify th
e l
eft and
the right Korn-Lichtenstein equations, namely
G
11
/G
22
,
G
22
/G
11
and
g
11
/g
22
,
g
22
/g
11
,
respectively. Indeed, by transforming
left and right,
KL 1st and KL 2nd are satisfied. Second, we analyze the left and the right Laplace-Beltrami
equations as the integrability conditions of the left and the right Korn-Lichtenstein equations,
namely by
{
Q
Λ
,Q
Φ
,P
Λ
,P
Φ
}
as well as
{
q
λ
,q
φ
,p
λ
,p
φ
}
A
1
,A
1
,A
2
r
, respec-
{
G
11
,G
22
,P
Λ
,P
Φ
,Q
Λ
,Q
Φ
}
as well as
{
g
11
,g
22
,p
λ
,p
φ
,q
λ
,q
φ
}
of
E
and
S
tively. Finally, we succeed to prove that
are
right
harmonic coordinates
. Third, we prove left and right orientation by computing the left and right
Jacobians which are notably positive.
{
P,Q
}
are
left harmonic coordinates
and
{
p, q
}
End of Solution (the first problem).
Solution (the second problem).
Again, we have to refer to standard textbooks of Differential Geometry, where you will find the
representation of the Gaussian curvature of a surface in terms of conformal coordinates (isometric,
isothermal). Let the left and the right matrix of the metric be equipped with a conformally flat
structure
{
G
l
=
λ
l
I
2
,
G
r
=
λ
r
I
2
}
, which is generated by a left and a right conformal coordinate
representation. Then the left Gaussian curvature and the right Gaussian curvature are provided
by
k
l
=
−
(1
/
2
λ
l
)
Δ
l
ln
λ
l
=
−
(1
/λ
l
)
Δ
l
ln
λ
l
and
k
r
=
−
(1
/
2
λ
r
)
Δ
r
ln
λ
r
=
−
(1
/λ
r
)
Δ
r
ln
λ
r
as well
as
Δ
l
:=
D
PP
+
D
QQ
=
D
P
+
D
Q
and
D
p
+
D
q
=
D
pp
+
D
qq
:=
Δ
r
,where
Δ
l
and
Δ
r
represent
the left Laplace-Beltrami operator and the right Laplace-Beltrami operator. Let us apply this
result in solving the second problem again.
•
By means of Box
1.34
, we have outlined how to generate a conformally flat metric of an ellipsoid-
of-revolution and of a sphere. First, we depart from the arc lengths “left d
S
2
” given in “left
coordinates”
as well as “right d
s
2
” given in “right coordinates”
. Second, we
compute the left and right Cauchy-Green matrices from “left d
P
2
+d
Q
2
”and “right d
p
2
+d
q
2
,
the arc lengths squared of the projective plane covered by left conformal coordinates
{
Λ, Φ
}
{
λ, φ
}
{
P, Q
}
and by right conformal coordinates
, respectively. In particular, we arrive at the two
equations d
P
2
+d
Q
2
=
f
2
(
Φ
)d
Λ
2
+
f
2
(
Φ
)d
Φ
2
and d
p
2
+d
q
2
=
g
2
(
φ
)d
λ
2
+
g
2
(
φ
)d
φ
2
,and
the corresponding elements of the left Cauchy-Green matrix C
l
and of the right Cauchy-
Green matrix C
r
. Third, we determine the left eigenvalues
{
p, q
}
l
Λ
1
,lΛ
2
}
and the right eigenvalues
{
r
Λ
l
,
r
Λ
2
}
in solving the left characteristic equation
|
C
l
−Λ
l
G
l
|
= 0 and the right characteristic
equation
|
C
r
−Λ
r
G
r
|
= 0. In particular, we prove the identities “left
l
Λ
1
=
l
Λ
2
=
Λ
l
”and“right
r
Λ
1
=
r
Λ
2
=
Λ
r
”, characteristic for a conformal mapping. Fourth, due to the duality relations
Λ
l
λ
l
=1
Λ
r
λ
r
= 1, we are able to compute
λ
l
and
λ
r
, respectively, and the conformally flat
metric of type “left d
S
2
=
λ
l
(d
P
2
+d
Q
2
)” and of type “right d
s
2
=
λ
r
(d
p
2
+d
q
2
)”.
{
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