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f
2
−gl
+
fm fl− em
.
1
K=
(1.295)
−gm
+
fnfm− en
eg
−
l
with respect to the basis
{
E
1
,
E
2
,
E
3
, |O}
fixed to the origin
O
and assumed to be orthonormal.
N
=
E
1
N
1
+
E
1
N
2
+
E
3
N
3
and
X
I
(
U, V
) are the representers of
Φ
−
l
, which are also called
embedding functions
N
I
denotes the coordinates of the surface normal vector
N
∈ N
M
M
l
⊂
R
3
l
. The “Theorem Egregium” of
C. F. Gauss states that the determinant of the curvature matrix, in short
Gaussian curvature
,
depends only on (i) the metric coecients
e, f, g
, (ii) their first derivatives
e
U
,e
V
,f
U
,f
V
,g
U
,g
V
,
and (iii) their second derivatives
e
UU
,e
UV
,e
VV
,...,g
UU
,g
UV
,g
VV
. The fundamental theorem of
an isometric mapping can now be formulated as follows.
if we exclude
self-intersections
and
singular points
(corners) of
M
Theorem 1.17 (Isometric mapping).
If a left curvature is isometrically mapped to a right surface, then corresponding points
X
∈
M
l
and
x
∈
M
r
have identical Gaussian curvature.
End of Theorem.
A list of Gaussian curvatures for different surfaces is shown in Table
1.5
. In consequence, there
are no isometries (i) from ellipsoid to sphere, (ii) from ellipsoid or sphere to plane, cylinder, cone,
any ruled surface (developable surfaces of Gaussian curvature zero).
Tab l e 1 . 5
Gaussian curvatures for some surfaces
Type of surface
Gaussian curvature
k
=
R
2
>
0
Sphere
S
2
R
A
1
(1
−E
2
)
E
2
A
1
,A
1
,A
2
k
=
MN
,M
:=
√
1
−E
2
sin
2
Φ
A
1
Ellipsoid-of-revolution
(1
−E
2
sin
2
Φ
)
3
/
2
,N
:=
Plane, cylinder, cone, ruled surface
k
=0
1-142 Equidistant Mapping of Submanifolds
Indeed, we are unable to produce an isometric landscape of the Earth, its Moon, the Sun, and
planets, other celestial bodies, or the universe. In this situation, we have to look for a softer
version of a length preserving mapping. Such an alternative concept is found by “dimension
reduction”. Only a one-dimensional
submanifold
M
1
of the two-dimensional
Riemann manifold
M
2
is mapped “length preserving”. For instance, we map the left coordinate line “ellipsoidal
equator” equidistantly to the right coordinate line “spherical equator”, namely by the postulate
A
1
Λ
=
rλ
. The arc length
A
1
Λ
of the ellipsoidal equator coincides with the arc length of the
spherical equator
rλ
. A more precise definition is given in Definition
1.18
.
Definition 1.18 (Equidistant mapping).
r
of a left surface (left two-dimensional Riemann manifold)
to a right surface (right two-dimensional Riemann manifold) be given. Beside the exceptional
points, both parameterized surfaces
Let a particular mapping
f
:
M
l
→
M
M
l
as well as
M
r
are covered by a set of coordinate lines
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