74
1 From Riemann manifolds to Riemann manifolds
1-11 Areal distortion
“It isn’t that they can’t see the solution. It is that they can’t see the problem.”
(G. K. Chesterton, The Scandal of Father Brown. The Point of a Pin.)
Fourth multiplicative and additive measures of deformation, dual deformation measures, areomorphism,
equiareal mapping.
Up to now, all deformation measures have been built on the first differential invariants I l and I r of
surface geometry, which are also called dS
2 and ds
2 . Such an invariant “left” or “right” measures
the infinitesimal distance between two points on the “left” or the “right” surface. A dual measure of
a surface (two-dimensional Riemann manifold) immersed in R
3 is the infinitesimal surface element.
Indeed, the surface element “left versus right”, (1.264), is dual to the infinitesimal distance element
“left versus right”, (1.265):
dS l :=
det[G l ]dU ∧ dV versus dS r :=
det[G r ]du ∧ dv ,
(1.264)
dS
2 = G 11 dU
2 + 2G 12 dU dV + G 22 dV
2 versus ds
2 = g 11 du
2 + 2g 12 dudv + g 22 dv
2 .
(1.265)
In the context of the mapping f : M
2
l → M
2
r , we next define areomorphism as an equiareal mapping
M
2
l → M
2
r : see Definition 1.13.
Definition 1.13 (Equiareal mapping).
An orientation preserving diffeomorphism f : M
2
l → M
2
r is called area preserving and equiareal (vector
product preserving, areomorphism) if
det[G l ]dU ∧ dV =
det[G r ]du ∧ dv
(1.266)
or, equivalently,
Φ
2
l = Φ
2 :=
det[G r ]du ∧ dv
det[G l ]dU ∧ dV
= 1
⇔
1 =
det[G l ]dU ∧ dV
det[G r ]du ∧ dv
=: Φ
2 = Φ
2
r
(1.267)
or
S lr :=
det[G r ]du ∧ dv −
det[G l ]dU ∧ dV = 0
⇔
S rl :=
det[G l ]dU ∧ dV −
det[G r ]du ∧ dv = 0
(1.268)
for all points of M
2
l and M
2
r , respectively, holds.
End of Definition.
Indeed, the left surface element
det[G l ]dU ∧ dV as well as the right surface element
det[G r ]du ∧ dv
have enabled us to introduce dual measures to the left length element dU
T G l dU as well as to the right
length element du
T G r du. There exist representations of the multiplicative measure of areal distortion,
{Φ
2
l , Φ
2
r }, and of the additive measure of areal distortion, {S lr , S rl }, in terms of the Cauchy–Green
deformation tensor, the Euler–Lagrange deformation tensor, and the principal stretches (left or right
eigenvalues), which we collect in Box 1.39 and turn out to be useful in the equivalence theorem.
1 From Riemann manifolds to Riemann manifolds
1-11 Areal distortion
“It isn’t that they can’t see the solution. It is that they can’t see the problem.”
(G. K. Chesterton, The Scandal of Father Brown. The Point of a Pin.)
Fourth multiplicative and additive measures of deformation, dual deformation measures, areomorphism,
equiareal mapping.
Up to now, all deformation measures have been built on the first differential invariants I l and I r of
surface geometry, which are also called dS
2 and ds
2 . Such an invariant “left” or “right” measures
the infinitesimal distance between two points on the “left” or the “right” surface. A dual measure of
a surface (two-dimensional Riemann manifold) immersed in R
3 is the infinitesimal surface element.
Indeed, the surface element “left versus right”, (1.264), is dual to the infinitesimal distance element
“left versus right”, (1.265):
dS l :=
det[G l ]dU ∧ dV versus dS r :=
det[G r ]du ∧ dv ,
(1.264)
dS
2 = G 11 dU
2 + 2G 12 dU dV + G 22 dV
2 versus ds
2 = g 11 du
2 + 2g 12 dudv + g 22 dv
2 .
(1.265)
In the context of the mapping f : M
2
l → M
2
r , we next define areomorphism as an equiareal mapping
M
2
l → M
2
r : see Definition 1.13.
Definition 1.13 (Equiareal mapping).
An orientation preserving diffeomorphism f : M
2
l → M
2
r is called area preserving and equiareal (vector
product preserving, areomorphism) if
det[G l ]dU ∧ dV =
det[G r ]du ∧ dv
(1.266)
or, equivalently,
Φ
2
l = Φ
2 :=
det[G r ]du ∧ dv
det[G l ]dU ∧ dV
= 1
⇔
1 =
det[G l ]dU ∧ dV
det[G r ]du ∧ dv
=: Φ
2 = Φ
2
r
(1.267)
or
S lr :=
det[G r ]du ∧ dv −
det[G l ]dU ∧ dV = 0
⇔
S rl :=
det[G l ]dU ∧ dV −
det[G r ]du ∧ dv = 0
(1.268)
for all points of M
2
l and M
2
r , respectively, holds.
End of Definition.
Indeed, the left surface element
det[G l ]dU ∧ dV as well as the right surface element
det[G r ]du ∧ dv
have enabled us to introduce dual measures to the left length element dU
T G l dU as well as to the right
length element du
T G r du. There exist representations of the multiplicative measure of areal distortion,
{Φ
2
l , Φ
2
r }, and of the additive measure of areal distortion, {S lr , S rl }, in terms of the Cauchy–Green
deformation tensor, the Euler–Lagrange deformation tensor, and the principal stretches (left or right
eigenvalues), which we collect in Box 1.39 and turn out to be useful in the equivalence theorem.