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In-Depth Information
2-3 The Equivalence Theorem for Conformal Mappings
The equivalence theorem for conformal mappings from the left two-dimensional Riemann
manifold to the right two-dimensional Euclidean manifold (conformeomorphism), Korn-
Lichtenstein equations and Cauchy-Riemann equations (d'Alembert-Euler equations).
The previous equivalence theorem for a conformeomorphism is specialized for the case of the
two-dimensional right Euclidean manifold
{
M
r
,g
μν
}
=
{
R
2
,δ
μν
}
=:
E
2
. In many applications,
2
,δ
μν
}
l
of the left
the choice of
{
R
is the planar manifold, for instance, the tangent space
T
U
o
M
two-dimensional Riemann manifold fixed to the point
U
0
=
, being covered by Cartesian
or polar coordinates. For an illustration of such a setup of a “planar manifold”, go back to our
previous examples.
{
U
0
,U
0
}
2-31 Conformeomorphism
First, let us confront you with Lemma
2.3
. The proof based upon Theorem
1.11
is straightforward.
Examples are given in the following chapters.
2
2
,δ
μν
Lemma 2.3 (Conformeomorphism, conformal mapping, special case
{
M
r
,g
μν
}
=
{
R
}
).
2
2
,δ
μν
Let
f
:
M
l
→{
R
}
be an orientation preserving conformal mapping. Then the following
conditions are equivalent.
(i)
Ψ
l
(
˙
1
,
˙
U
U
2
)=
Ψ
r
( ˙
u
1
,
˙
u
2
)
(2.18)
˙
1
,
˙
for all tangent vectors
U
U
2
and their images ˙
u
1
,
˙
u
2
, respectively.
(ii) C
l
=
λ
2
(
U
0
)G
l
versus C
r
=
λ
2
I
2
,
C
−
r
=I
2
/λ
2
,
C
11
=
C
22
=
λ
2
, C
12
=
C
21
=0
, C
11
=
C
22
=
λ
−
2
, C
12
=
C
21
=0;
(2.19)
E
l
=
K
(
U
0
)G
l
versus E
r
=
κ
I
2
,
E
−
r
=I
2
/κ,
E
11
=
E
22
=
κ, E
12
=
E
21
=0
, E
11
=
E
22
=
κ
−
1
, E
12
=
E
21
=0
.
(iii)
K
=(
Λ
2
(
λ
2
,
−
1)
/
2
Λ
2
=2
K
+1
−
1)
/
2=
κ
2
κ
+1=
λ
2
versus
Λ
1
=
Λ
2
=
Λ
(
U
0
)versus
λ
1
=
λ
2
=
λ
(
u
0
)
,
(2.20)
K
1
=
K
2
=
K
(
U
0
) s
κ
1
=
κ
2
=
κ
(
u
0
)
,
Λ
2
(
U
0
) = tr[C
l
G
−
l
]
/
2 s
λ
2
(
u
0
) = tr[C
r
]
/
2;
(left dilatation)
K
= tr[E
l
G
−
l
]
/
2
versus (right dilatation)
κ
= tr[E
r
]
/
2
,
tr[C
l
G
−
l
]=2
det [C
l
G
−
l
]
tr[C
l
G
−
l
]=2
det [C
r
]
,
versus
(2.21)
tr[E
l
G
−
l
]=2
det [E
l
G
−
l
]
tr[E
r
]=2
det [E
r
]
.
versus
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