80
1 From Riemann manifolds to Riemann manifolds
1-14 Review: the canonical criteria
“Where we cannot use the compass of mathematics or the torch of experience
. . . it is certain we cannot take a single step forward.”
(Voltaire.)
Review: the canonical criteria for conformal equiareal, isometric, and equidistant mappings, optimal map
projections, Gaussian curvatures.
Up to now, we have defined the conformal mapping (compare with Definition 1.10) as well as the
equiareal mapping (compare with Definition 1.13) from the left two-dimensional Riemann manifold
(here: left surface immersed into R
3 ) to the right two-dimensional Riemann manifold (here: right
surface immersed into R
3 ). We demonstrated that under the action of the conformal map, angles were
preserved. In contrast, an equiareal transformation preserves the surface element. However, what is
to tell about length preserving mappings f : M
2
l → M
2
r ?
1-141 Isometry
Let us begin with the definition of an isometry and relate it in the form of an equivalence theorem
to the other measures of deformation. In particular, we ask the question: When does an isometric
mapping exist?
Definition 1.15 (Isometry).
An admissible mapping f : M
2
l → M
2
r is called length preserving or an isometry if for any curve in the
left surface (“left curve”: c l (t l ), t l ∈ I(c l )) the corresponding curve in the right surface (“right curve”:
c r (t r ), t r ∈ I(c r )) as its image f ◦ c l (t l ) has the identical length:
b l
a l
˙
s l dt l =
b r
a r
˙
s r dt r .
(1.289)
Two Riemann manifolds M
2
l and M
2
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 M
2
r .)
Theorem 1.16 (Isometry M
2
l → M
2
r ).
An admissible mapping f : M
2
l → M
2
r is an isometry if and only if the following equivalent 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)
(ii) The stretches Λ for any point X ∈ M
1
l ⊂ M
2
l ⊂ R
3 is independent of the directions of
the tangent vector ˙
X, a constant to be one, i. e.
Λ( ˙
X) = 1 ∀ ˙
X = 0 , ˙
X ∈ T M
1
l ⊂ T M
2
l , ˙
X =
3
I=1
2
M =1
E I
∂X
I
∂U M
dU
M
dt l
.
(1.291)
(iii) The left principal stretches for any point X ∈ M
2
l are a constant to be one: Λ 1 = Λ 2 = 1.
End of Theorem.
1 From Riemann manifolds to Riemann manifolds
1-14 Review: the canonical criteria
“Where we cannot use the compass of mathematics or the torch of experience
. . . it is certain we cannot take a single step forward.”
(Voltaire.)
Review: the canonical criteria for conformal equiareal, isometric, and equidistant mappings, optimal map
projections, Gaussian curvatures.
Up to now, we have defined the conformal mapping (compare with Definition 1.10) as well as the
equiareal mapping (compare with Definition 1.13) from the left two-dimensional Riemann manifold
(here: left surface immersed into R
3 ) to the right two-dimensional Riemann manifold (here: right
surface immersed into R
3 ). We demonstrated that under the action of the conformal map, angles were
preserved. In contrast, an equiareal transformation preserves the surface element. However, what is
to tell about length preserving mappings f : M
2
l → M
2
r ?
1-141 Isometry
Let us begin with the definition of an isometry and relate it in the form of an equivalence theorem
to the other measures of deformation. In particular, we ask the question: When does an isometric
mapping exist?
Definition 1.15 (Isometry).
An admissible mapping f : M
2
l → M
2
r is called length preserving or an isometry if for any curve in the
left surface (“left curve”: c l (t l ), t l ∈ I(c l )) the corresponding curve in the right surface (“right curve”:
c r (t r ), t r ∈ I(c r )) as its image f ◦ c l (t l ) has the identical length:
b l
a l
˙
s l dt l =
b r
a r
˙
s r dt r .
(1.289)
Two Riemann manifolds M
2
l and M
2
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 M
2
r .)
Theorem 1.16 (Isometry M
2
l → M
2
r ).
An admissible mapping f : M
2
l → M
2
r is an isometry if and only if the following equivalent 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)
(ii) The stretches Λ for any point X ∈ M
1
l ⊂ M
2
l ⊂ R
3 is independent of the directions of
the tangent vector ˙
X, a constant to be one, i. e.
Λ( ˙
X) = 1 ∀ ˙
X = 0 , ˙
X ∈ T M
1
l ⊂ T M
2
l , ˙
X =
3
I=1
2
M =1
E I
∂X
I
∂U M
dU
M
dt l
.
(1.291)
(iii) The left principal stretches for any point X ∈ M
2
l are a constant to be one: Λ 1 = Λ 2 = 1.
End of Theorem.