Geography Reference
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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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