82
1 From Riemann manifolds to Riemann manifolds
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).
Let a particular mapping f : M
2
l → M
2
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 M
2
l as well as M
2
r are covered by a set of coordinate lines {U = constant, V },
{U, V = constant} as well as {u = constant, v}, {u, v = constant}, called left curves c l (t) (left
one-dimensional submanifold) and right curves c r (t) (right one-dimensional submanifold). Under the
mapping f ◦ c l (t) = c r (t), the mapping
c l (t) → c r (t) or R
3
⊃ M
2
l ⊃ M
1
l
equidistant
−→
M
1
r ⊂ M
2
r ⊂ R
3
(1.296)
is called equidistant if a finite section of a specific left curve c l (t) has the same length as a finite section
of a corresponding right curve c r (t).
End of Definition.
Let us work out the equivalence theorem for an equidistant mapping from a left curve c l (t) to a right
curve c r (t).
Theorem 1.19 (Equidistant mapping R
3
⊃ M
2
l ⊃ M
1
l −→M
1
r ⊂ M
2
r ⊂ R
3 ).
Let us assume that the left surface (left two-dimensional Riemann manifold) as well as the right surface
(right two-dimensional Riemann manifold) has been parameterized by left coordinates {U, V } and
right coordinates {u, v}. If the directions of their left tangent vectors and their right tangent vectors
coincide with the directions of the left principal stretches (left eigendirections, left eigenvectors) and
of the right principal stretches (right eigendirections, right eigenvectors), then the following conditions
of an equidistant mapping are equivalent.
(i) Equidistant mapping of a section of a specific left curve c l (t) to a
corresponding section of a specific right curve c r (t).
U coordinate line to u coordinate line:
V coordinate line to v coordinate line:
b l
a l
G 22 (t) ˙
V dt =
b r
a r
g 22 (t) ˙
vdt .
b l
a l
G 11 (t) ˙
U dt =
b r
a r
g 11 (t) ˙
udt .
(1.297)
(ii) Left or right Cauchy–Green matrix under an equidistant mapping c l (t) → c r (t).
U coordinate line to u coordinate line:
V coordinate line to v coordinate line:
c 22 = G 22 or C 22 = g 22 .
c 11 = G 11 or C 11 = g 11 .
(1.298)
(iii) Left or right principal stretches under an equidistant mapping c l (t) → c r (t).
U coordinate line to u coordinate line:
V coordinate line to v coordinate line:
Λ 2 = 1 or λ 2 = 1 .
Λ 1 = 1 or λ 1 = 1 .
(1.299)
End of Theorem.
1 From Riemann manifolds to Riemann manifolds
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).
Let a particular mapping f : M
2
l → M
2
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 M
2
l as well as M
2
r are covered by a set of coordinate lines {U = constant, V },
{U, V = constant} as well as {u = constant, v}, {u, v = constant}, called left curves c l (t) (left
one-dimensional submanifold) and right curves c r (t) (right one-dimensional submanifold). Under the
mapping f ◦ c l (t) = c r (t), the mapping
c l (t) → c r (t) or R
3
⊃ M
2
l ⊃ M
1
l
equidistant
−→
M
1
r ⊂ M
2
r ⊂ R
3
(1.296)
is called equidistant if a finite section of a specific left curve c l (t) has the same length as a finite section
of a corresponding right curve c r (t).
End of Definition.
Let us work out the equivalence theorem for an equidistant mapping from a left curve c l (t) to a right
curve c r (t).
Theorem 1.19 (Equidistant mapping R
3
⊃ M
2
l ⊃ M
1
l −→M
1
r ⊂ M
2
r ⊂ R
3 ).
Let us assume that the left surface (left two-dimensional Riemann manifold) as well as the right surface
(right two-dimensional Riemann manifold) has been parameterized by left coordinates {U, V } and
right coordinates {u, v}. If the directions of their left tangent vectors and their right tangent vectors
coincide with the directions of the left principal stretches (left eigendirections, left eigenvectors) and
of the right principal stretches (right eigendirections, right eigenvectors), then the following conditions
of an equidistant mapping are equivalent.
(i) Equidistant mapping of a section of a specific left curve c l (t) to a
corresponding section of a specific right curve c r (t).
U coordinate line to u coordinate line:
V coordinate line to v coordinate line:
b l
a l
G 22 (t) ˙
V dt =
b r
a r
g 22 (t) ˙
vdt .
b l
a l
G 11 (t) ˙
U dt =
b r
a r
g 11 (t) ˙
udt .
(1.297)
(ii) Left or right Cauchy–Green matrix under an equidistant mapping c l (t) → c r (t).
U coordinate line to u coordinate line:
V coordinate line to v coordinate line:
c 22 = G 22 or C 22 = g 22 .
c 11 = G 11 or C 11 = g 11 .
(1.298)
(iii) Left or right principal stretches under an equidistant mapping c l (t) → c r (t).
U coordinate line to u coordinate line:
V coordinate line to v coordinate line:
Λ 2 = 1 or λ 2 = 1 .
Λ 1 = 1 or λ 1 = 1 .
(1.299)
End of Theorem.