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(
t
):
π
2(1 + cos2
t
)d
t
=
1
−E
2
×
ln
1+
E
2
E
1
−E
+
1
−
E
sin
Φ
1+
E
sin
Φ
E
cos
Φ
E
sin
Φ
+
E
cos
Φ
(1 +
E
sin
Φ
)
×
1
−
(1
−
E
sin
Φ
)
2
(1
− E
2
sin
2
Φ
)
2
d
Φ,
1+cos2
t
=2cos
2
t,
1
− E
2
sin
2
Φ
+
4
E
3
sin
2
Φ
cos
Φ
2
E
cos
Φ
+
(2.63)
E
cos
Φ
π
cos
2
t
(
t
)=
,
E
2
sin
2
Φ
)
2
ln
1+
E
2
E
1
−E
2
(1
−
1
−E
+
E
2
cos
2
Φ
π
2
cos
4
t
(
t
)
2
=
1
−E
2
2
,
ln
1+
E
E
2
sin
2
Φ
)
4
(1
−
2
E
1
−E
+
ln
1+
E
1
E
2
2
(1
− E
2
sin
2
Φ
)
4
A
1
cos
2
ΦA
1
(1
1
G
11
G
22
=
E
)
2
,
2
b
2
=
A
1
(1
−
E
2
)
2
2
E
E
+
.
−
π
2
E
2
−
1
−
(6th) Determinantal identity:
det [C
l
G
−
l
]=1
.
(2.64)
2-5 Canonical Criteria for Conformal, Equiareal, and Other Mappings
Canonical criteria for conformal, equiareal, andisometricmappingsaswellasequidistant
mappings
l
2
,δ
μν
}
M
→{
R
, Hilbert invariants.
Question: “How can we generalize those canonical crite-
ria for a conformal, an equiareal, or an isometric map-
ping
2
if we restrict the right
two-dimensional Riemann manifold to be two-dimensional
Euclidean?” Answer: “Let us refer to Boxes
1.46
and
1.47
in
order to formulate the answer. As it is outlined in Box
2.6
,
the fundamental four Hilbert invariants
I
1
and
I
2
or
i
1
and
i
2
become dependent, typically called 'syzygetic', as soon as
we are dealing with a conformal mapping
2
l
2
2
,δ
μν
}
M
→
M
r
:=
{
R
=
E
2
l
2
,δ
μν
M
→{
R
}
.”
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