Geography Reference
In-Depth Information
Two Riemann manifolds
M
l
and
M
r
. respectively, which are mapped on each other by means of
an isometry are called
isometric
.
End of Definition.
Without any proof, we make the following equivalence statement. (Of course, we could make an
equivalent statement for the right manifold
2
M
r
.)
l
r
).
Theorem 1.16 (Isometry
M
→
M
2
l
M
r
is an isometry if and only if the following equivalent
An admissible mapping
f
:
M
→
conditions are fulfilled.
(i) The coordinates of the left Cauchy-Green tensor C
l
are identical to the coordinates of the left metric tensor G
l
,
i.e.
C
l
=G
l
.
(1.290)
1
l
2
l
3
is independent
(ii) The stretches
Λ
for any point
X
∈
M
⊂
M
⊂
R
X
,
a constant to be one, i.e.
of the directions of the tangent vector
3
2
E
I
∂X
I
∂U
M
d
U
M
d
t
l
Λ
(
X
)=1
∀
X
=0
,
X
∈ T
M
X
=
1
l
2
⊂ T
M
l
,
.
(1.291)
I
=1
M
=1
2
l
(iii) The left principal stretches for any point
X
∈
M
are a constant to be one:
Λ
1
=
Λ
2
=1
.
End of Theorem.
r
were existing for an arbitrary left and right two-
dimensional Riemann manif
ol
d, we would have met an ideal situation. Let us therefore ask: when
does an isometric mapping
f
:
M
If an isometric mapping
f
:
M
l
→
M
r
exist? Unfortunately, we can only sketch the existence
proof here which is based upon the intrinsic measure of curvature of a surface, namely Gaussian
curvature, computed
i
→
M
k
=det[K]=
det[H]
det[G]
,
(1.292)
HG
−
1
2
×
2
.
K:=
−
∈
R
The curvature matrix K of a surface is the negative product of the Hesse matrix H and the inverse
of the Gauss matrix G defined as follows.
Box 1.45 (Curvature matrix of a surface).
∂U
N
=
ef
,
3
∂X
I
∂U
M
∂X
I
G=
(1.293)
fg
I
=1
∂U
M
∂U
N
N
l
=
lm
,
3
∂
2
X
I
H=
(1.294)
mn
I
=1
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