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Definition 1.10 (Conformal mapping).
2
l
2
r
is called
angle preserving
conformal
mapping (conformeomorphism, inner product preserving) if
Ψ
l
=
Ψ
r
and Σ
l
=
Σ
=0
An orientation preserving diffeomorphism
f
:
M
→
M
⇔
Σ
r
=
σ
=
2
2
0 for all points of
M
l
and
M
r
, respectively, holds.
End of Definition.
2
l
2
Theorem 1.11 (Conformeomorphism
M
→
M
r
, conformal mapping).
2
l
2
Let
f
:
r
be an orientation preserving conformal mapping. Then the following conditions
(i)-(iv) are equivalent:
M
→
M
(i)
Ψ
l
(
U
1
, U
2
)=
Ψ
r
(
u
1
, u
2
)
,
(1.153)
U
1
, U
2
,
respectively;
(ii) C
l
=
Λ
2
(
U
0
)G
l
,
C
l
G
−
l
=
Λ
2
(
U
0
)I
2
versus C
r
=
λ
2
(
u
0
)G
r
,
C
r
G
−
r
=
λ
2
(
u
0
)I
2
,
(1.154)
for all tangent vectors
{
}
and their images
{
u
1
, u
2
}
E
l
=
K
(
U
0
)G
l
,
E
l
G
−
l
=
K
(
U
0
)I
2
versus E
r
=
κ
(
u
0
)G
r
,
E
r
G
−
r
=
κ
(
u
0
)I
2
;
(iii)
K
=(
Λ
2
−
1)
/
2=
κ,
2
κ
+1=
λ
2
,
Λ
1
=
Λ
2
=
Λ
(
U
0
) sus
λ
1
=
λ
2
=
λ
(
u
0
)
,
K
1
=
K
2
=
K
(
U
0
)versus
κ
1
=
κ
2
=
κ
(
u
0
)
,
−
1)
/
2
,Λ
2
=2
K
+1versus(
λ
2
(1.155)
Λ
2
(
U
0
)=
1
2
tr[C
l
G
−
i
] sus
λ
2
(
u
0
)=
1
2
tr[C
r
G
−
r
];
left dilatation:
right dilatation:
K
=
1
2
tr[E
l
G
−
l
] sus
κ
=
1
2
tr[E
r
G
−
r
]
,
tr[C
l
G
−
l
]=2
det[C
l
G
−
l
] versus tr[C
r
G
−
r
]=2
det[C
r
G
−
r
]
,
(1.156)
tr[E
l
G
−
l
]=2
det[E
l
G
−
l
] versus tr[E
r
G
−
r
]=2
det[E
r
G
−
r
]
,
(iv) generalized Korn-Lichtenstein equations (special case:
g
12
=0):
u
U
u
V
=
g
11
g
22
−G
12
G
11
−G
22
G
12
v
U
v
V
,
1
G
11
G
22
(1.157)
G
12
−
subject to the integrability conditions
u
UV
=
u
VU
and
v
UV
=
v
VU
.
End of Theorem.
Before we present the sketches of proofs for the various conditions, it has to be noted that the
generalized Korn-Lichtenstein equations, which govern conformal mapping
r
,sufferfrom
the defect that they contain the unknown functions
g
11
[
u
λ
(
U
A
)] and
g
22
[
u
λ
(
U
Λ
)], and the reason
is that the mapping functions
u
λ
(
U
Λ
) have to be determined. In case of
2
l
M
→
M
r
,g
μν
2
,δ
μν
,
the corresponding Korn-Lichtenstein equations do not suffer since these functions do not appear.
The stated problem is overcome by representing the right Riemann manifold
M
{
M
}
=
{
R
}
r
by
isometric
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